ID: 842 Type: Subjective Source: NCERT Class 11 Year: null

Question 1: Find the limits: $\lim_{x\rightarrow1} \left[x^{3}-x^{2}+1\right]$

ID: 843 Type: Subjective Source: NCERT Class 11 Year: null

Question 2: Find the limits: $\lim_{x\rightarrow3} \left[x(x+1)\right]$

ID: 844 Type: Subjective Source: NCERT Class 11 Year: null

Question 3: Find the limits: $\lim_{x\rightarrow-1} \left[1+x+x^{2}+...+x^{10}\right]$

ID: 845 Type: Subjective Source: NCERT Class 11 Year: null

Question 4: Find the limits: $\lim_{x\rightarrow1} \left[\frac{x^{2}+1}{x+100}\right]$

ID: 846 Type: Subjective Source: NCERT Class 11 Year: null

Question 5: Find the limits: $\lim_{x\rightarrow2} \left[\frac{x^{3}-4x^{2}+4x}{x^{2}-4}\right]$

ID: 847 Type: Subjective Source: NCERT Class 11 Year: null

Question 6: Find the limits: $\lim_{x\rightarrow2} \left[\frac{x^{2}-4}{x^{3}-4x^{2}+4x}\right]$

ID: 848 Type: Subjective Source: NCERT Class 11 Year: null

Question 7: Find the limits: $\lim_{x\rightarrow2} \left[\frac{x^{3}-2x^{2}}{x^{2}-5x+6}\right]$

ID: 849 Type: Subjective Source: NCERT Class 11 Year: null

Question 8: Find the limits: $\lim_{x\rightarrow1} \left[\frac{x-2}{x^{2}-x}-\frac{1}{x^{3}-3x^{2}+2x}\right]$

ID: 850 Type: Subjective Source: NCERT Class 11 Year: null

Question 9: Evaluate: $\lim_{x\rightarrow1}\frac{x^{15}-1}{x^{10}-1}$

ID: 851 Type: Subjective Source: NCERT Class 11 Year: null

Question 10: Evaluate: $\lim_{x\rightarrow0}\frac{\sqrt{1+x}-1}{x}$

ID: 852 Type: Subjective Source: NCERT Class 11 Year: null

Question 11: Evaluate: $\lim_{x\rightarrow0}\frac{\sin 4x}{\sin 2x}$

ID: 853 Type: Subjective Source: NCERT Class 11 Year: null

Question 12: Evaluate: $\lim_{x\rightarrow0}\frac{\tan x}{x}$

ID: 854 Type: Subjective Source: NCERT Class 11 Year: null

Question 13: Evaluate the following limits . $\lim_{x\rightarrow3} x+3$

ID: 855 Type: Subjective Source: NCERT Class 11 Year: null

Question 14: Evaluate the following limits . $\lim_{x\rightarrow\pi} (x-\frac{22}{7})$

ID: 856 Type: Subjective Source: NCERT Class 11 Year: null

Question 15: Evaluate the following limits . $\lim_{r\rightarrow1} \pi r^{2}$

ID: 857 Type: Subjective Source: NCERT Class 11 Year: null

Question 16: Evaluate the following limits . $\lim_{x\rightarrow4}\frac{4x+3}{x-2}$

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🤖 AI Generated Solution:

To evaluate the limit, we can substitute $x=4$ directly into the expression since the denominator is non-zero at $x=4$. Step 1: Substitute $x=4$ into the expression. $$ \lim_{x\rightarrow4}\frac{4x+3}{x-2} = \frac{4(4)+3}{4-2} $$ Step 2: Simplify the numerator and the denominator. $$ \frac{16+3}{2} = \frac{19}{2} $$ The limit is $\frac{19}{2}$. The final answer is $\boxed{\frac{19}{2}}$.

ID: 858 Type: Subjective Source: NCERT Class 11 Year: null

Question 17: Evaluate the following limits . $\lim_{x\rightarrow-1}\frac{x^{10}+x^{5}+1}{x-1}$

ID: 859 Type: Subjective Source: NCERT Class 11 Year: null

Question 18: Evaluate the following limits . $\lim_{x\rightarrow0}\frac{(x+1)^{5}-1}{x}$

ID: 860 Type: Subjective Source: NCERT Class 11 Year: null

Question 19: Evaluate the following limits . $\lim_{x\rightarrow2}\frac{3x^{2}-x-10}{x^{2}-4}$

ID: 861 Type: Subjective Source: NCERT Class 11 Year: null

Question 20: Evaluate the following limits . $\lim_{x\rightarrow3}\frac{x^{4}-81}{2x^{2}-5x-3}$

ID: 862 Type: Subjective Source: NCERT Class 11 Year: null

Question 21: Evaluate the following limits . $\lim_{x\rightarrow0}\frac{ax+b}{cx+1}$

ID: 863 Type: Subjective Source: NCERT Class 11 Year: null

Question 22: Evaluate the following limits . $\lim_{z\rightarrow1}\frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}$

ID: 864 Type: Subjective Source: NCERT Class 11 Year: null

Question 23: Evaluate the following limits . $\lim_{x\rightarrow1}\frac{ax^{2}+bx+c}{cx^{2}+bx+a}$, $a+b+c\ne0$

ID: 865 Type: Subjective Source: NCERT Class 11 Year: null

Question 24: Evaluate the following limits . $\lim_{x\rightarrow-2}\frac{\frac{1}{x}+\frac{1}{2}}{x+2}$

ID: 866 Type: Subjective Source: NCERT Class 11 Year: null

Question 25: Evaluate the following limits . $\lim_{x\rightarrow0}\frac{\sin ax}{bx}$

ID: 867 Type: Subjective Source: NCERT Class 11 Year: null

Question 26: Evaluate the following limits . $\lim_{x\rightarrow0}\frac{\sin ax}{\sin bx}$, $a, b\ne0$

ID: 868 Type: Subjective Source: NCERT Class 11 Year: null

Question 27: Evaluate the following limits . $\lim_{x\rightarrow\pi}\frac{\sin (\pi-x)}{\pi(\pi-x)}$

ID: 869 Type: Subjective Source: NCERT Class 11 Year: null

Question 28: Evaluate the following limits . $\lim_{x\rightarrow0}\frac{\cos x}{\pi-x}$

ID: 870 Type: Subjective Source: NCERT Class 11 Year: null

Question 29: Evaluate the following limits . $\lim_{x\rightarrow0}\frac{\cos 2x-1}{\cos x-1}$

ID: 871 Type: Subjective Source: NCERT Class 11 Year: null

Question 30: Evaluate the following limits . $\lim_{x\rightarrow0}\frac{ax+x\cos x}{b\sin x}$

ID: 872 Type: Subjective Source: NCERT Class 11 Year: null

Question 31: Evaluate the following limits . $\lim_{x\rightarrow0} x\sec x$

ID: 873 Type: Subjective Source: NCERT Class 11 Year: null

Question 32: Evaluate the following limits . $\lim_{x\rightarrow0}\frac{\sin ax+bx}{ax+\sin bx}$, $a, b, a+b\ne0$

ID: 874 Type: Subjective Source: NCERT Class 11 Year: null

Question 33: Evaluate the following limits . $\lim_{x\rightarrow0}(\operatorname{cosec} x-\cot x)$

ID: 875 Type: Subjective Source: NCERT Class 11 Year: null

Question 34: Evaluate the following limits . $\lim_{x\rightarrow\frac{\pi}{2}}\frac{\tan 2x}{x-\frac{\pi}{2}}$

ID: 876 Type: Subjective Source: NCERT Class 11 Year: null

Question 35: Find $\lim_{x\rightarrow0}f(x)$ and $\lim_{x\rightarrow1}f(x)$ where $f(x)=\begin{cases} 2x+3,&x\le0\\ 3(x+1),&x>0\end{cases}$

ID: 877 Type: Subjective Source: NCERT Class 11 Year: null

Question 36: Find $\lim_{x\rightarrow1}f(x)$ where $f(x)=\begin{cases} x^{2}-1,&x\le1\\ -x^{2}-1,&x>1\end{cases}$

ID: 878 Type: Subjective Source: NCERT Class 11 Year: null

Question 37: Evaluate $\lim_{x\rightarrow0}f(x)$ where $f(x)=\begin{cases}\frac{|x|}{x},&x\ne0\\ 0,&x=0\end{cases}$

ID: 879 Type: Subjective Source: NCERT Class 11 Year: null

Question 38: Find $\lim_{x\rightarrow0}f(x)$ where $f(x)=\begin{cases}\frac{x}{|x|^{\circ}},&x\ne0\\ 0,&x=0\end{cases}$

ID: 880 Type: Subjective Source: NCERT Class 11 Year: null

Question 39: Find $\lim_{x\rightarrow5}f(x)$ where $f(x)=|x|-5$

ID: 881 Type: Subjective Source: NCERT Class 11 Year: null

Question 40: Suppose $f(x)=\begin{cases}a+bx,&x<1\\ 4,&x=1\\ b-ax,&x>1\end{cases}$ and $\lim_{x\rightarrow1}f(x)=f(1)$. What are the possible values of $a$ and $b$?

ID: 882 Type: Subjective Source: NCERT Class 11 Year: null

Question 41: Let $a_{1},a_{2},...,a_{n}$ be fixed real numbers and define a function $f(x)=(x-a_{1})(x-a_{2})...(x-a_{n})$. What is $\lim_{x\rightarrow a_{1}}f(x)$? For some $a\ne a_{1},a_{2},...,a_{n}$, compute $\lim_{x\rightarrow a}f(x)$.

ID: 883 Type: Subjective Source: NCERT Class 11 Year: null

Question 42: If $f(x)=\begin{cases}|x|+1,&x<0\\ 0,&x=0\\ |x|-1,&x>0\end{cases}$. For what value(s) of $a$ does $\lim_{x\rightarrow a}f(x)$ exist?

ID: 884 Type: Subjective Source: NCERT Class 11 Year: null

Question 43: If the function $f(x)$ satisfies $\lim_{x\rightarrow1}\frac{f(x)-2}{x^{2}-1}=\pi$, evaluate $\lim_{x\rightarrow1}f(x)$.

ID: 885 Type: Subjective Source: NCERT Class 11 Year: null

Question 44: If $f(x)=\begin{cases}mx^{2}+n,&x<0\\ nx+m,&0\le x\le1\\ nx^{3}+m,&x>1\end{cases}$, for what integers $m$ and $n$ does both $\lim_{x\rightarrow0}f(x)$ and $\lim_{x\rightarrow1}f(x)$ exist?

ID: 886 Type: Subjective Source: NCERT Class 11 Year: null

Question 45: Find the derivative at $x=2$ of the function $f(x)=3x$

ID: 887 Type: Subjective Source: NCERT Class 11 Year: null

Question 46: Find the derivative of the function $f(x)=2x^{2}+3x-5$ at $x=-1$. Also prove that $f'(0)+3f'(-1)=0$.

ID: 888 Type: Subjective Source: NCERT Class 11 Year: null

Question 47: Find the derivative of $\sin x$ at $x=0$.

ID: 889 Type: Subjective Source: NCERT Class 11 Year: null

Question 48: Find the derivative of $f(x)=3$ at $x=0$ and at $x=3$.

ID: 890 Type: Subjective Source: NCERT Class 11 Year: null

Question 49: Find the derivative of $f(x)=10x$.

ID: 891 Type: Subjective Source: NCERT Class 11 Year: null

Question 50: Find the derivative of $f(x)=x^{2}$.

ID: 892 Type: Subjective Source: NCERT Class 11 Year: null

Question 51: Find the derivative of the constant function $f(x)=a$ (a number $a$).

ID: 893 Type: Subjective Source: NCERT Class 11 Year: null

Question 52: Find the derivative of $f(x)=\frac{1}{x}$.

ID: 894 Type: Subjective Source: NCERT Class 11 Year: null

Question 53: Compute the derivative of $6x^{100}-x^{55}+x$.

ID: 895 Type: Subjective Source: NCERT Class 11 Year: null

Question 54: Find the derivative of $f(x)=1+x+x^{2}+x^{3}+...+x^{50}$ at $x=1$.

ID: 896 Type: Subjective Source: NCERT Class 11 Year: null

Question 55: Find the derivative of $f(x)=\frac{x+1}{x}$.

ID: 897 Type: Subjective Source: NCERT Class 11 Year: null

Question 56: Compute the derivative of $\sin x$.

ID: 898 Type: Subjective Source: NCERT Class 11 Year: null

Question 57: Compute the derivative of $f(x)=\tan x$.

ID: 899 Type: Subjective Source: NCERT Class 11 Year: null

Question 58: Compute the derivative of $f(x)=\sin^{2}x$.

ID: 900 Type: Subjective Source: NCERT Class 11 Year: null

Question 59: Find the derivative of $f$ from the first principle, where $f$ is given by $f(x)=\frac{2x+3}{x-2}$

ID: 901 Type: Subjective Source: NCERT Class 11 Year: null

Question 60: Find the derivative of $f$ from the first principle, where $f$ is given by $f(x)=x+\frac{1}{x}$

ID: 902 Type: Subjective Source: NCERT Class 11 Year: null

Question 61: Find the derivative of $f(x)=\sin x+\cos x$ from the first principle.

ID: 903 Type: Subjective Source: NCERT Class 11 Year: null

Question 62: Find the derivative of $f(x)=x\sin x$ from the first principle.

ID: 904 Type: Subjective Source: NCERT Class 11 Year: null

Question 63: Compute derivative of $f(x)=\sin 2x$.

ID: 905 Type: Subjective Source: NCERT Class 11 Year: null

Question 64: Compute derivative of $g(x)=\cot x$.

ID: 906 Type: Subjective Source: NCERT Class 11 Year: null

Question 65: Find the derivative of $\frac{x^{5}-\cos x}{\sin x}$.

ID: 907 Type: Subjective Source: NCERT Class 11 Year: null

Question 66: Find the derivative of $\frac{x+\cos x}{\tan x}$.

ID: 908 Type: Subjective Source: NCERT Class 11 Year: null

Question 67: Find the derivative of $x^{2}-2$ at $x=10$.

ID: 909 Type: Subjective Source: NCERT Class 11 Year: null

Question 68: Find the derivative of $x$ at $x=1$.

ID: 910 Type: Subjective Source: NCERT Class 11 Year: null

Question 69: Find the derivative of $\frac{1}{x}$ at $x=100$.

ID: 911 Type: Subjective Source: NCERT Class 11 Year: null

Question 70: Find the derivative of the following functions from first principle: $(x-1)(x-2)$

ID: 912 Type: Subjective Source: NCERT Class 11 Year: null

Question 71: Find the derivative of the following functions from first principle: $x^{3}-27$

ID: 913 Type: Subjective Source: NCERT Class 11 Year: null

Question 72: Find the derivative of the following functions from first principle: $\frac{1}{x^{2}}$

ID: 914 Type: Subjective Source: NCERT Class 11 Year: null

Question 73: Find the derivative of the following functions from first principle: $\frac{x+1}{x-1}$

ID: 915 Type: Subjective Source: NCERT Class 11 Year: null

Question 74: For the function $f(x)=\frac{x^{100}}{100}+\frac{x^{99}}{99}+...+\frac{x^{2}}{2}+x+1$. Prove that $f'(1)=100f'(0)$.

ID: 916 Type: Subjective Source: NCERT Class 11 Year: null

Question 75: Find the derivative of $x^{n}+ax^{n-1}+a^{2}x^{n-2}+...+a^{n-1}x+a^{n}$ for some fixed real number $a$.

ID: 917 Type: Subjective Source: NCERT Class 11 Year: null

Question 76: For some constants $a$ and $b$, find the derivative of $(x-a)(x-b)$.

ID: 918 Type: Subjective Source: NCERT Class 11 Year: null

Question 77: For some constants $a$ and $b$, find the derivative of $(ax^{2}+b)^{2}$.

ID: 919 Type: Subjective Source: NCERT Class 11 Year: null

Question 78: For some constants $a$ and $b$, find the derivative of $\frac{x-a}{x-b}$.

ID: 920 Type: Subjective Source: NCERT Class 11 Year: null

Question 79: Find the derivative of $\frac{x^{n}-a^{n}}{x-a}$ for some constant $a$.

ID: 921 Type: Subjective Source: NCERT Class 11 Year: null

Question 80: Find the derivative of $2x-\frac{3}{4}$.

ID: 922 Type: Subjective Source: NCERT Class 11 Year: null

Question 81: Find the derivative of $(5x^{3}+3x-1)(x-1)$.

ID: 923 Type: Subjective Source: NCERT Class 11 Year: null

Question 82: Find the derivative of $x^{-3}(5+3x)$.

ID: 924 Type: Subjective Source: NCERT Class 11 Year: null

Question 83: Find the derivative of $x^{5}(3-6x^{-9})$.

ID: 925 Type: Subjective Source: NCERT Class 11 Year: null

Question 84: Find the derivative of $x^{-4}(3-4x^{-5})$.

ID: 926 Type: Subjective Source: NCERT Class 11 Year: null

Question 85: Find the derivative of $\frac{2}{x+1}-\frac{x^{2}}{3x-1}$.

ID: 927 Type: Subjective Source: NCERT Class 11 Year: null

Question 86: Find the derivative of $\cos x$ from first principle.

ID: 928 Type: Subjective Source: NCERT Class 11 Year: null

Question 87: Find the derivative of the following functions: $\sin x \cos x$

ID: 929 Type: Subjective Source: NCERT Class 11 Year: null

Question 88: Find the derivative of the following functions: $2\tan x-7\sec x$

ID: 930 Type: Subjective Source: NCERT Class 11 Year: null

Question 89: Find the derivative of the following functions: $5\sec x+4\cos x$

ID: 931 Type: Subjective Source: NCERT Class 11 Year: null

Question 90: Find the derivative of the following functions: $\operatorname{cosec} x$

ID: 932 Type: Subjective Source: NCERT Class 11 Year: null

Question 91: Find the derivative of the following functions: $3\cot x+5\operatorname{cosec} x$

ID: 933 Type: Subjective Source: NCERT Class 11 Year: null

Question 92: Find the derivative of the following functions: $5\sin x-6\cos x+7$

ID: 934 Type: Subjective Source: NCERT Class 11 Year: null

Question 93: Find the derivative of the following functions from first principle: $-x$

ID: 935 Type: Subjective Source: NCERT Class 11 Year: null

Question 94: Find the derivative of the following functions from first principle: $(-x)^{-1}$

ID: 936 Type: Subjective Source: NCERT Class 11 Year: null

Question 95: Find the derivative of the following functions from first principle: $\sin(x+1)$

ID: 937 Type: Subjective Source: NCERT Class 11 Year: null

Question 96: Find the derivative of the following functions from first principle: $\cos(x-\frac{\pi}{8})$

ID: 938 Type: Subjective Source: NCERT Class 11 Year: null

Question 97: Find the derivative of $(x+a)$.

ID: 939 Type: Subjective Source: NCERT Class 11 Year: null

Question 98: Find the derivative of $(px+q)(\frac{r}{x}+s)$.

ID: 940 Type: Subjective Source: NCERT Class 11 Year: null

Question 99: Find the derivative of $(ax+b)(cx+d)^{2}$.

ID: 941 Type: Subjective Source: NCERT Class 11 Year: null

Question 100: Find the derivative of $\frac{ax+b}{cx+d}$.

ID: 942 Type: Subjective Source: NCERT Class 11 Year: null

Question 101: Find the derivative of $\frac{1+\frac{1}{x}}{1-\frac{1}{x}}$.

ID: 943 Type: Subjective Source: NCERT Class 11 Year: null

Question 102: Find the derivative of $\frac{1}{ax^{2}+bx+c}$.

ID: 944 Type: Subjective Source: NCERT Class 11 Year: null

Question 103: Find the derivative of $\frac{ax+b}{px^{2}+qx+r}$.

ID: 945 Type: Subjective Source: NCERT Class 11 Year: null

Question 104: Find the derivative of $\frac{px^{2}+qx+r}{ax+b}$.

ID: 946 Type: Subjective Source: NCERT Class 11 Year: null

Question 105: Find the derivative of $\frac{a}{x^{4}}-\frac{b}{x^{2}}+\cos x$.

ID: 947 Type: Subjective Source: NCERT Class 11 Year: null

Question 106: Find the derivative of $4\sqrt{x}-2$.

ID: 948 Type: Subjective Source: NCERT Class 11 Year: null

Question 107: Find the derivative of $(ax+b)^{n}$.

ID: 949 Type: Subjective Source: NCERT Class 11 Year: null

Question 108: Find the derivative of $(ax+b)^{n}(cx+d)^{m}$.

ID: 950 Type: Subjective Source: NCERT Class 11 Year: null

Question 109: Find the derivative of $\sin(x+a)$.

ID: 951 Type: Subjective Source: NCERT Class 11 Year: null

Question 110: Find the derivative of $\operatorname{cosec} x \cot x$.

ID: 952 Type: Subjective Source: NCERT Class 11 Year: null

Question 111: Find the derivative of $\frac{\cos x}{1+\sin x}$.

ID: 953 Type: Subjective Source: NCERT Class 11 Year: null

Question 112: Find the derivative of $\frac{\sin x+\cos x}{\sin x-\cos x}$.

ID: 954 Type: Subjective Source: NCERT Class 11 Year: null

Question 113: Find the derivative of $\frac{\sec x-1}{\sec x+1}$.

ID: 955 Type: Subjective Source: NCERT Class 11 Year: null

Question 114: Find the derivative of $\sin^{n}x$.

ID: 956 Type: Subjective Source: NCERT Class 11 Year: null

Question 115: Find the derivative of $\frac{a+b\sin x}{c+d\cos x}$.

ID: 957 Type: Subjective Source: NCERT Class 11 Year: null

Question 116: Find the derivative of $\frac{\sin(x+a)}{\cos x}$.

ID: 958 Type: Subjective Source: NCERT Class 11 Year: null

Question 117: Find the derivative of $x^{4}(5\sin x-3\cos x)$.

ID: 959 Type: Subjective Source: NCERT Class 11 Year: null

Question 118: Find the derivative of $(x^{2}+1)\cos x$.

ID: 960 Type: Subjective Source: NCERT Class 11 Year: null

Question 119: Find the derivative of $(ax^{2}+\sin x)(p+q\cos x)$.

ID: 961 Type: Subjective Source: NCERT Class 11 Year: null

Question 120: Find the derivative of $(x+\cos x)(x-\tan x)$.

ID: 962 Type: Subjective Source: NCERT Class 11 Year: null

Question 121: Find the derivative of $\frac{4x+5\sin x}{3x+7\cos x}$.

ID: 963 Type: Subjective Source: NCERT Class 11 Year: null

Question 122: Find the derivative of $\frac{x^{2}\cos(\frac{\pi}{4})}{\sin x}$.

ID: 964 Type: Subjective Source: NCERT Class 11 Year: null

Question 123: Find the derivative of $\frac{x}{1+\tan x}$.

ID: 965 Type: Subjective Source: NCERT Class 11 Year: null

Question 124: Find the derivative of $(x+\sec x)(x-\tan x)$.

ID: 966 Type: Subjective Source: NCERT Class 11 Year: null

Question 125: Find the derivative of $\frac{x}{\sin^{n}x}$.

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🤖 AI Generated Solution:

We need to find the derivative of the function $f(x) = \frac{x}{\sin^n x}$. We will use the quotient rule, which states that if $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$. Here, $u(x) = x$ and $v(x) = \sin^n x$. Step 1: Find the derivative of $u(x)$. $u'(x) = \frac{d}{dx}(x) = 1$. Step 2: Find the derivative of $v(x)$. We use the chain rule here. Let $y = \sin^n x$. Let $z = \sin x$. Then $y = z^n$. $\frac{dy}{dz} = nz^{n-1}$ and $\frac{dz}{dx} = \cos x$. So, $v'(x) = \frac{dy}{dx} = \frac{dy}{dz} \cdot \frac{dz}{dx} = n(\sin x)^{n-1} \cdot \cos x = n \sin^{n-1} x \cos x$. Step 3: Apply the quotient rule. $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$ $f'(x) = \frac{(1)(\sin^n x) - (x)(n \sin^{n-1} x \cos x)}{(\sin^n x)^2}$ Step 4: Simplify the expression. $f'(x) = \frac{\sin^n x - nx \sin^{n-1} x \cos x}{\sin^{2n} x}$ We can divide each term in the numerator by $\sin^{2n} x$. $f'(x) = \frac{\sin^n x}{\sin^{2n} x} - \frac{nx \sin^{n-1} x \cos x}{\sin^{2n} x}$ $f'(x) = \sin^{n-2n} x - nx \sin^{n-1-2n} x \cos x$ $f'(x) = \sin^{-n} x - nx \sin^{-n-1} x \cos x$ We can also factor out $\sin^{-n-1} x$: $f'(x) = \sin^{-n-1} x (\sin x - nx \cos x)$ Alternatively, we can rewrite $\sin^{-n} x$ as $\frac{1}{\sin^n x}$: $f'(x) = \frac{1}{\sin^n x} - \frac{nx \cos x}{\sin^{n+1} x}$ To combine these terms, find a common denominator, which is $\sin^{n+1} x$: $f'(x) = \frac{\sin x}{\sin^{n+1} x} - \frac{nx \cos x}{\sin^{n+1} x}$ $f'(x) = \frac{\sin x - nx \cos x}{\sin^{n+1} x}$ The final answer is $\boxed{\frac{\sin x - nx \cos x}{\sin^{n+1} x}}$.