JEE Main Differentiation Practice Questions With Solutions

If ${\log }_{e}y=3{\sin }^{-1}x$log_{e} ⁡ y = 3 sin^{- 1} ⁡ x, then $(1-x^2)y^{\prime \prime }-x{y}^{\prime }$\left(\right. 1 - x^{2} \left.\right) y^{′ ′} - x y^{′} at $x=\frac{1}{2}$x = \frac{1}{2} is equal to

Let $f(x)=a{x}^3+b{x}^2+cx+41$f \left(\right. x \left.\right) = a x^{3} + b x^{2} + c x + 41 be such that $f(1)=40,f^{\prime }(1)=2$f \left(\right. 1 \left.\right) = 40 , f^{′} \left(\right. 1 \left.\right) = 2 and $f^{\prime \prime }(1)=4$f^{′ ′} \left(\right. 1 \left.\right) = 4. Then $a^2+b^2+c^2$a^{2} + b^{2} + c^{2} is equal to:

Suppose for a differentiable function $h,h(0)=0,h(1)=1$h , h \left(\right. 0 \left.\right) = 0 , h \left(\right. 1 \left.\right) = 1 and $h^{\prime }(0)=h^{\prime }(1)=2$h^{′} \left(\right. 0 \left.\right) = h^{′} \left(\right. 1 \left.\right) = 2. If $g(x)=h\left({\mathrm{e}}^x\right){\mathrm{e}}^{h(x)}$g \left(\right. x \left.\right) = h \left(\right. e^{x} \left.\right) e^{h \left(\right. x \left.\right)}, then $g^{\prime }(0)$g^{′} \left(\right. 0 \left.\right) is equal to:

$\text{ If }f(x)=\{\begin{array}{ll}{x}^3\sin \left(\frac{1}{x}\right),& x\ne 0\\ 0& ,x=0\end{array}\text{, then }$\textrm{ }\text{If}\textrm{ } f \left(\right. x \left.\right) = \left{\right. x^{3} sin ⁡ \left(\right. \frac{1}{x} \left.\right) , & x \neq 0 \\ 0 & , x = 0 ,\text{ then}\textrm{ }

Let $f:(-\mathrm{\infty },\mathrm{\infty })-\{0\}\to \mathbb{R}$f : \left(\right. - \infty , \infty \left.\right) - \left{\right. 0 \left.\right} \rightarrow \mathbb{R} be a differentiable function such that $f^{\prime }(1)=\underset{a\to \mathrm{\infty }}{lim}{a}^{2}f\left(\frac{1}{a}\right)$f^{′} \left(\right. 1 \left.\right) = \underset{a \rightarrow \infty}{lim} a^{2} f \left(\right. \frac{1}{a} \left.\right). Then $\underset{a\to \mathrm{\infty }}{lim}\frac{a(a+1)}{2}{\tan }^{-1}\left(\frac{1}{a}\right)+a^2-2{\log }_{e}a$\underset{a \rightarrow \infty}{lim} \frac{a \left(\right. a + 1 \left.\right)}{2} tan^{- 1} ⁡ \left(\right. \frac{1}{a} \left.\right) + a^{2} - 2 log_{e} ⁡ a is equal to

If $y(\theta )=\frac{2\cos \theta +\cos 2\theta }{\cos 3\theta +4\cos 2\theta +5\cos \theta +2}$y \left(\right. \theta \left.\right) = \frac{2 cos ⁡ \theta + cos ⁡ 2 \theta}{cos ⁡ 3 \theta + 4 cos ⁡ 2 \theta + 5 cos ⁡ \theta + 2}, then at $\theta =\frac{\pi }{2},y^{\prime \prime }+y^{\prime }+y$\theta = \frac{\pi}{2} , y^{′ ′} + y^{′} + y is equal to :

Let $f(x)=x^5+2{\mathrm{e}}^{x/4}$f \left(\right. x \left.\right) = x^{5} + 2 e^{x / 4} for all $x\in \mathbf{R}$x \in \mathbf{R}. Consider a function $g(x)$g \left(\right. x \left.\right) such that $(g\circ f)(x)=x$\left(\right. g \circ f \left.\right) \left(\right. x \left.\right) = x for all $x\in \mathbf{R}$x \in \mathbf{R}. Then the value of $8g^{\prime }(2)$8 g^{′} \left(\right. 2 \left.\right) is :

Let $f:\mathbb{R}-\{0\}\to \mathbb{R}$f : \mathbb{R} - \left{\right. 0 \left.\right} \rightarrow \mathbb{R} be a function satisfying $f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$f \left(\right. \frac{x}{y} \left.\right) = \frac{f \left(\right. x \left.\right)}{f \left(\right. y \left.\right)} for all $x,y,f(y)\ne 0$x , y , f \left(\right. y \left.\right) \neq 0. If $f^{\prime }(1)=2024$f^{′} \left(\right. 1 \left.\right) = 2024, then

Let $g:\mathbf{R}\to \mathbf{R}$g : \mathbf{R} \rightarrow \mathbf{R} be a non constant twice differentiable function such that ${\mathrm{g}}^{\prime }\left(\frac{1}{2}\right)={\mathrm{g}}^{\prime }\left(\frac{3}{2}\right)$g^{′} \left(\right. \frac{1}{2} \left.\right) = g^{′} \left(\right. \frac{3}{2} \left.\right). If a real valued function $f$f is defined as $f(x)=\frac{1}{2}[g(x)+g(2-x)]$f \left(\right. x \left.\right) = \frac{1}{2} \left[\right. g \left(\right. x \left.\right) + g \left(\right. 2 - x \left.\right) \left]\right., then

If $f(x)=\left|\begin{array}{ccc}2{\cos }^{4}x& 2{\sin }^{4}x& 3+{\sin }^{2}2x\\ 3+2{\cos }^{4}x& 2{\sin }^{4}x& {\sin }^{2}2x\\ 2{\cos }^{4}x& 3+2{\sin }^{4}x& {\sin }^{2}2x\end{array}\right|,$f \left(\right. x \left.\right) = \left|\right. 2 cos^{4} ⁡ x & 2 sin^{4} ⁡ x & 3 + sin^{2} ⁡ 2 x \\ 3 + 2 cos^{4} ⁡ x & 2 sin^{4} ⁡ x & sin^{2} ⁡ 2 x \\ 2 cos^{4} ⁡ x & 3 + 2 sin^{4} ⁡ x & sin^{2} ⁡ 2 x \left|\right. , then $\frac{1}{5}{f}^{\prime }(0)=$\frac{1}{5} f^{′} \left(\right. 0 \left.\right) = is equal to :