List of top Mathematics Questions on Differentiation asked in JEE Main

Let \( f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f^{(3)}(3) \) for all \( x \in \mathbb{R} \). Then the value of \( f'(5) \) is:

Let \(f\) be a differentiable function satisfying \[ f(x)=1-2x+\int_0^x (t-x)f(t)\,dt,\quad x\in\mathbb{R}, \] and let \[ g(x)=\int_0^x \{f(t)+2\}^5(t-4)^6(t+12)^7\,dt. \] If \(p\) and \(q\) are respectively the points of local minima and local maxima of \(g\), then the value of \(|p+q|\) is _______.

If \(6\int_{1}^{x} f(t)dt = 3x f(x) + x^3 - 4, x \geq 1\) then value of (f(2)-f(3)) is :

Let \( L_1 : \frac{x - 1}{3} = \frac{y}{4} = \frac{z}{5} \) and \( L_2 : \frac{x - p}{2} = \frac{y}{3} = \frac{z}{4} \). If the shortest distance between \( L_1 \) and \( L_2 \) is \( \frac{1}{\sqrt{6}} \), then the possible value of \( p \) is:

Let \( f : \mathbb{R} \to \mathbb{R} \) be a thrice differentiable function such that \[ f(0) = 0, \, f(1) = 1, \, f(2) = -1, \, f(3) = 2, \, \text{and} \, f(4) = -2. \] Then, the minimum number of zeros of \( (3f' f' + f'') (x) \) is:

Let \( f(x) = x^5 + 2e^{x/4} \) for all \( x \in \mathbb{R} \). Consider a function \( g(x) \) such that \( (g \circ f)(x) = x \) for all \( x \in \mathbb{R} \). Then the value of \( 8g'(2) \) is:

Suppose\(f(x)=\frac{(2^x+2^{-x})tanx\sqrt{tan^{-1}(x^2-x+1)}}{(7x^2+3x+1)^{3}}\) then the value of \(f'(0)\) is equal to

Let \( g(x) = 3f\left(\frac{x}{3}\right) + f(3 - x) \) and \( f''(x) >0 \) for all \( x \in (0, 3) \). If \( g \) is decreasing in \( (0, \alpha) \) and increasing in \( (\alpha, 3) \), then \( 8\alpha \) is:

Let \(f(x) = \cos\left(2\tan^{-1} \sin \left(\cot^{-1} \sqrt{\frac{1-x}{x}}\right)\right)\), \(0<x<1\). Then :

If $y = y(x)$ is an implicit function of $x$ such that $\log_e (x + y) = 4xy$, then $\frac{d^2y}{dx^2}$ at $x = 0$ is equal to _________.