Algebra of Derivative of Functions

if y = y(x) and it follows the relation $e^{xy}+ycosx=\mathrm{2,}$ then find (i) y' (0) and (ii) y'' (0)

Let $\text{f},\text{g}$ and $\text{h}$ are differentiable functions. If $\text{f}\left(0\right)=1;\text{g}\left(0\right)=2;\text{h}\left(0\right)=3$ and the derivatives of their pair wise products at $\text{x}=0$ are ${\left(\text{fg}\right)}^{\text{'}}\left(0\right)=6\text{;}{\left(\text{gh}\right)}^{\text{'}}\left(0\right)=4$ and ${\left(\text{hf}\right)}^{\text{'}}\left(0\right)=5$, then the value of ${\left(\text{fgh}\right)}^{\text{'}}\left(0\right)$ is

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

Find $\frac{dy}{dx}$ in the following case

$x^3{y}^4=x+y$

Let $f\left(x\right)=\sum _{k=1}^{10}k·x^k, x\in \mathrm{ℝ}$, if $2f\left(2\right)+f^{\text{'}}\left(2\right)=119{\left(2\right)}^n+1$ then $n$ is equal to ______.

Differentiate the following w.r.t $x$.

$y=x^7\sin x-a^x\cos x$

If $y={\cos }^{-1}\left(\frac{x^{2n}-1}{x^{2n}+1}\right)$, then $\left(1+x^{2n}\right)y_1$ is

The value of $\frac{d\left[x^n\left({\log }_{a}x\right)e^x\right]}{dx}$ is

Find the derivative of $x^2{a}^{2x}$ w.r.t. $\text{'}x\text{'}$.

Let $f\left(x\right)=\left(x-1\right)\left(x-2\right)\left(x-3\right)$, then prove that there is more than one $c$ in $\left(1,3\right)$ such that $f^{\text{'}}\left(c\right)=0$.

Find the derivative of $y=x(5\sin x-3\cos x)$

Find the derivative of $\left(\frac{x-1}{x}\right)\left(\frac{x^2-1}{x^2}\right)$ with respect to $x.$

If $f\left(x\right)=\sqrt{\frac{1+x^2}{1-x^2}}; \left|x\right|<1$, then find $f^{\text{'}}\left(x\right)$.

Let $f\left(x\right)=\left(x+1\right)\left(x-2\right)\left(x+3\right)\left(x-4\right)\left(x+5\right)\left(x-6\right)\left(x+7\right)\left(x-8\right)\left(x+9\right)\left(x-10\right)$, then value of $\frac{f\text{'}\left(-5\right)}{8!}$ is equal to

If $y=\frac{3\sin x+\cos x+4}{\cos x}$, then find $\frac{dy}{dx}$ at $x=\frac{\mathrm{\pi }}{4}$.

Find the derivative of $\frac{x+\cos x}{\tan x}$.

If $y=e^x·x^2\cos x$, then $\frac{dy}{dx}$ equals

Differentiate the function $-\frac{x}{y\sqrt{y^2-x^2}}$.

If $f\left(x\right)=\frac{\sqrt{1+x^2}}{\sqrt{1-x^2}}$ the find $f\text{'}\left(x\right)$