Question 1:
If \(f(x)=-2x^{8}\) then the correct statement is :
Question 2:
The derivative of \(\sin(x^{2})\) w.r.t. x, at \(x=\sqrt{\pi}\) is :
Question 3:
The derivative of \(\tan^{-1}(x^{2})\) w.r.t. x is :
Question 4:
The derivative of \(2^{x}\) w.r.t. \(3^{x}\) is:
Question 5:
Derivative of \(e^{2x}\) with respect to \(e^{x}\), is:
Question 6:
If \(xe^{y}=1\), then the value of \(\frac{dy}{dx}\) at \(x=1\) is :
Question 7:
Derivative of \(e^{\sin^{2}x}\) with respect to cos x is:
Question 8:
Differentiate \(2^{\cos^2 x}\) w.r.t \(\cos^2 x\)
Question 9:
If \(\tan^{-1} \left(x^2 + y^2\right) = a^2\), then find \(\frac{dy}{dx}\)
Question 10:
If \(x=e^{\frac{x}{y}}\), then prove that \(\frac{dy}{dx}=\frac{x-y}{x~\log x}\).
Question 11:
Differentiate \(\frac{\sin x}{\sqrt{\cos x}}\) with respect to \(x\).
Question 12:
If \(y=5\cos x-3\sin x\), prove that \(\frac{d^{2}y}{dx^{2}}+y=0.\)
Question 13:
Differentiate \(\sqrt{e^{\sqrt{2x}}}\) with respect to \(e^{\sqrt{2x}}\) for \(x>0.\)
Question 14:
If \((x)^{y}=(y)^{x}\), then find \(\frac{dy}{dx}\).
Question 15:
Differentiate \((\frac{5^{x}}{x^{5}})\) with respect to x.
Question 16:
If \(-2x^{2}-5xy+y^{3}=76\), then find \(\frac{dy}{dx}\).
Question 17:
If \(f(x)=|\tan~2x|\), then find the value of \(f^{\prime}(x)\) at \(x=\frac{\pi}{3}\)
Question 18:
If \(x=e^{x/y}\), prove that \(\frac{dy}{dx}=\frac{\log~x-1}{(\log~x)^{2}}\)
Question 19:
If \(y=\cos^{3}(\sec^{2}2t)\), find \(\frac{dy}{dt}\) .
Question 20:
If \(x^{y}=e^{x-y},\) prove that \(\frac{dy}{dx}=\frac{\log~x}{(1+\log~x)^{2}}.\)
Question 21:
If \(y=\operatorname{cosec}(\cot^{-1}x)\), then prove that \(\sqrt{1+x^{2}}\frac{dy}{dx}-x=0\)
Question 22:
If \(y=\sqrt{\tan\sqrt{x}}\), prove that \(\sqrt{x}\frac{dy}{dx}=\frac{1+y^{4}}{4y}\).
Question 23:
If \(x\sqrt{1+y}+y\sqrt{1+x}=0,\) with \( -1< x < 1,\), \(x\ne y,\) then prove that \(\frac{dy}{dx}=\frac{-1}{(1+x)^{2}}\).
Question 24:
Differentiate \(y=\sin^{-1}(3x-4x^{3})\) w.r.t. x, if \(x\in[-\frac{1}{2},\frac{1}{2}].\)
Question 25:
Differentiate \(y=\cos^{-1}(\frac{1-x^{2}}{1+x^{2}})\) with respect to x, when \(x\in(0,1).\)
Question 26:
Differentiate \(y=\sqrt{\log\{\sin(\frac{x^{3}}{3}-1)\}}\) with respect to x.
Question 27:
Find \(\frac{dy}{dx}\) , if \((cos~x)^{y}=(cos~y)^{x}\)
Question 28:
If \(\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y),\) prove that \(\frac{dy}{dx}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}.\)
Question 29:
If \(x~cos(p+y)+cos~p~sin(p+y)=0\), prove that \(cos~p\frac{dy}{dx}=-cos^{2}(p+y),\) where p is a constant.
Question 30:
If \(x=e^{\cos~3t}\) and \(y=e^{\sin~3t}\), prove that \(\frac{dy}{dx}=-\frac{y~\log~x}{x~\log~y}\)
Question 31:
If \(\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)\), prove that \(\frac{dy}{dx}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}.\)
Question 32:
If \(y=(\tan x)^{x},\) then find \(\frac{dy}{dx}\).
Question 33:
Show that: \(\frac{d}{dx}(|x|)=\frac{x}{|x|},x\ne0\)
Question 34:
Given that \(y=(sin~x)^{x}\cdot x^{sin~x}+a^{x},\) find \(\frac{dy}{dx}\)
Question 35:
If \(\sqrt{1-x^2} + \sqrt{1-y^2} = a(x-y)\), then prove that \(\frac{dy}{dx} = \sqrt{\frac{1-y^2}{1-x^2}}\).
Question 36:
For a positive constant 'a', differentiate \(a^{(t+\frac{1}{t})}\) with respect to \(t+\frac{1}{t}\) where t is a non-zero real number.
Question 37:
Find \(\frac{dy}{dx}\) if \(y^{x}+x^{y}+x^{x}=a^{b}\) where a and b are constants.