Recent questions tagged differentiation

Find $\dfrac{dy}{dx}$ where $$y=\int_{x^2}^{\sin x}\sqrt{1+t^4}dt$$$\cos x\sqrt{1+\sin^4x}-2x\sqrt{1+x^8}$ $\cos x\sqrt{1+\sin^4x}+2x\sqrt{1+x^8}$ $\sqrt{1+\sin^4x}-\sqrt...

Which of the following is the derivative of $f(x)=(x^2+1)^{x\ln x}$ when $x>0$?$(x^2+1)^{x\ln x}\left((\ln x+1)\ln(x^2+1)+\dfrac{2x^2\ln x}{x^2+1}\right)$ $(x^2+1)^{x\ln ...

The function $f(x)$ is defined by $$f(x)=\begin{cases}\dfrac{x^2+ax+b}{\sqrt{x+3}},&x\le 1\\\\ \ln(2x-1)+\dfrac{5x}{x+1},&x>1\end{cases}$$ where $a$ and $b$ are constants...

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as follows:\[f(x)=\left(\frac{|x|}{2}-x\right)\left(x-\frac{|x|}{2}\right)\]Which of the following statements is/are...

Let $m$ and $n$ be the numbers of points at which $f(x) = \max\{x, x^3, x^5, \dots, x^{21}\}$ for $x \in \mathbb{R}$ is not differentiable and not continuous, respectivel...

Let $f(x)=\sqrt{x}$. We draw a tangent to the curve $y=f(x)$ at the point on the curve whose $x$ coordinate is equal to $4$. Where does this tangent intersect the $X$-axi...

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function which is twice differentiable everywhere and suppose that $f^{\prime}(q)=0$ for every rational number $q.$ Find t...

For a positive integer $n$, define the function $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$ by $f_{n}(x)=$ $\sin \left(\frac{x}{n}\right)$. Consider the sequences $\left\{...

Suppose $f$ is a real-valued function defined on the set of real numbers. Let$$f(x)=\frac{\sin x}{x} \text { for } x \neq 0, \quad f(0)=a$$where $a$ is a real number. Let...

Suppose f is a real-valued function defined on the set of real numbers. Letf(x) = $\frac{sinx}{x}$ for x$\neq$0, f(0) = a, where a is a real number. Let f be differentiab...

Consider the given function $f(x)$.$$f(x)=\left\{\begin{array}{ll}a x+b & \text { for } x<1 \\x^{3}+x^{2}+1 & \text { for } x \geq 1\end{array}\right.$$If the function is...

If $y=\dfrac{4 x-5}{x+5},$ then $\dfrac{d y}{d x}$ equals$\frac{20}{(x+5)^2}$$\frac{25}{(x+5)^2}$$\frac{x+5}{4 x-5}$$4$