List of top Mathematics Questions on Continuity and differentiability asked in KEAM

Let $f(x)$ and $g(x)$ be two differentiable functions such that $f'(x)=g(x)$ and $g'(x)=-f(x)$. Let $h(x)=(f(x))^2+(g(x))^2$ and $h(3)=100$. Then $h(100)$ is equal to

If $(3 + 5x)e^{\frac{y}{x}} = x$, then $\dfrac{dy}{dx}$ is equal to:

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f(x)=x^3+x^2f'(1)+xf''(2)+f'''(3)$, then $f'''(3) =$

If $y=\cos x \cos y$, then $\frac{dy}{dx}$ at $\left(\frac{\pi}{3},\frac{\pi}{6}\right)$ is:

For $x\in\mathbb{R}$, let $f(x)=\log(3-\sin x)$ and $g(x)=f(f(x))$. Then $g'(0) =$

Let $f(x)=10-|x-5|,\; x\in\mathbb{R}$. Then $f(x)$ is not differentiable at:

If \( y = \tan^{-1}(x^2 - x) \), then \( \frac{dy}{dx} = \)

Let \( h(x) = f(\sqrt{g(x)}) \). If \( f'(3) = 6 \), \( g'(3) = 3 \) and \( g(3) = 9 \), then the value of \( h'(3) \) is equal to

If \( f(x) = |x^2 + x - 6| \) is not differentiable at \( x = a \) and \( x = b \), then \( a^2 + b^2 = \)

The domain of the function \( f(x) = \left(\sqrt{8x - x^2 - 7}\right)^{\frac{3}{2}} \) is:

Let \(f(x) = \frac{\sqrt[3]{x^4}}{\sqrt[3]{x^2}},\ x \neq 0\). Then the value of \(f'(27)\) is equal to

Let \(f(x) = \begin{cases} 5x^2 + ax + 16, & \text{if } x < 2 \\ x^2, & \text{if } x \geq 2 \end{cases}\). If \(f\) is differentiable at \(x = 2\), then the value of \(a\) is equal to

If \(f(x) = \frac{\sqrt{2\sin x}}{\sqrt{1 + \cos 2x}}\), then \(f'\left(\frac{\pi}{6}\right) =\)

Let \(f(x) = \frac{1}{x^2}\) and let \(u = f(x)f''(x)\). Then \(\frac{du}{dx} =\)

If \(e^{x}(y + 2\sqrt{1 + x}) = 5\), then \(-\frac{dy}{dx}\) at \((0,3)\) is

If \(y = \sec (\tan^{-1} x)\), then \(\frac{dy}{dx}\) at \(x = \sqrt{3}\) is equal to

If $\sqrt{\frac{y}{x}} + \sqrt{\frac{x}{y}} = 1$, then $\frac{dy}{dx}$ equals

Let \( f(x)=|x-2| \) and \( g(x)=f(f(x)) \). Then derivative of \( g \) at the point \( x=5 \) is

Let \( f:\mathbb{R} \to \mathbb{R} \) be a differentiable function. If \( f \) is even, then \( f'(0) \) is equal to

Let \( f:\mathbb{R} \to \mathbb{R} \) be a differentiable function. If \( f \) is even, then \( f'(0) \) is equal to

If \( y = \frac{\sin^2 x}{1+\cot x} + \frac{\cos^2 x}{1+\tan x} \), then \( y'(x) \) is equal to

Let \( f:(-1,1)\to(-1,1) \) be continuous, \( f(x)=f(x^2) \), and \( f(0)=\frac{1}{2} \). Find \( 4f\left(\frac{1}{4}\right) \)

Let \( f:\mathbb{R} \to \mathbb{R} \) be a differentiable function. If \( f \) is even, then \( f'(0) \) is equal to

If \( y = \frac{\sin^2 x}{1+\cot x} + \frac{\cos^2 x}{1+\tan x} \), then \( y'(x) \) is equal to