The Chain Rule

Calculate the derivative of the following functions.

$\displaystyle f(x)=\tan(\text{arccos}(e^{4x}))$

Find the derivative of the following function

$y=e^{\sin\sqrt{x}}$

Practice: Finding a Derivative

$\left(\tan^{-1}\left(3^{\sin x}\right)\right)'|_{x=\pi}=$

The derivative of $y=3^{e^x}$ is

If $f\left(x\right)=\left(g\left(x\right)-h\left(x\right)\right)^5$, $g\left(0\right)=2,\ h\left(0\right)=0,\ g'\left(0\right)=6$, and $h'\left(0\right)=-4$, find $f'\left(0\right)$.

Find $f'\left(1\right)$ if $f\left(x\right)=\sqrt{x^2\ g\left(x\right)}$, $g\left(1\right)=1$ and $g'\left(1\right)=2$.

Find the derivative of $f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}$

Practice: trig and inverse trig derivatives

Find the derivative of the following trigonometric and inverse trigonometric functions:

a) $f(x)=\arcsin(3x^2)$

Find $\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right)$.

Practice: Chain Rule

Find the derivative of $y=\ln(\arctan x)$

Find the derivative of the following function.

If $g\left(t\right)=\sqrt[3]{9+f(t)}$ , write an expression for the derivative $g'\left(t\right)$

Calculate the derivative of the function $f\left(x\right)=\sqrt{\tan\left(x^3\right)}$

Given the information $$$g(x)=\sqrt{1+[f(x)]^2} \ \text{ such that } f(1)=2,f'(1)=-3$. Find $g'(1).$ Express your answer as a fraction in lowest terms. If the answer is negative, put the negative sign in front of the entire fraction.

$\textbf{Q:}$ Given that $f(1)=5,\ g(1)=-2,\ f'(1)=2$ and $g'(1)=-1/2$, find the derivative of $\sqrt{f(x)-g^2(x)}$ at $x=1$.

$\textbf{Q.}$ Given that $g(1)=g^{\prime}(1)=1$, find $f^{\prime}(\pi/2)$ if $\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}$.

Find the derivative of $h(x)=f(g(x))$ if $$$f(x)=x^2+1$ and $g(x)=3\sqrt{x}$ .

$\bf{Q:}$ Find $(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'$

Find the derivative of the following function.

Find the derivative of the following function:

Find an expression for the derivative of the following function:

Find the derivative of the following function:

Compute the derivative of

Find $g'(0)$ for $g(x)=\sqrt{f(x^2)} + f(x),$ given $f(0)=2 \text{ and }f'(0)=1.$

Find the derivative of the following function.

Find the derivative of $h(x)=f(g(x))\,\, \text{ if }\, f(x)=x^2+1$ and $g(x)=3\sqrt{x}$.

Find $g^{\prime}(0)$ for $g(x)=\sqrt{f(x^2)}+f(x)$, given $f(0)=2$and $f^{\prime}(0)=1$.

Find the derivative of $h(x)=f(g(x))\,\, \text{ if }\, f(x)=x^2+1$ and $g(x)=3\sqrt{x}$.

If $2-g\left(x\right)=x^2+2\left[f\left(x\right)\right]^2-x^3g\left(x\right)$, $f\left(0\right)=1\ ,\ f'\left(0\right)=-2,\ g\left(0\right)=-1\ \text{and}\ \ g''\left(0\right)=3$, then the value of $f''\left(0\right)$ is equal to

The derivative of $y=\left(2-x^7\right)^{500}$ is

The derivative of $\log_3\left(e^{3x}\right)$ is

Find the derivative of the following functions at the given values

a) $f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1}$ at $x=0$

b) $g\left(x\right)=\left(\cos x\right)^{e^x}$ at $x=0$

Find the derivative of $h(x) = \log(\cos(x))$. Remember that $\log x = \log_e x = \ln x$.

Let $\displaystyle f(x) = \frac{\sqrt{x^2 - 6}}{x - 3}$.

$\textbf{Q:}$ Given that $f(1)=5,\ g(1)=-2,\ f'(1)=2$ and $g'(1)=-1/2$, find the derivative of $\sqrt{f(x)-g^2(x)}$ at $x=1$.

$\textbf{Q.}$ Given that $g(1)=g^{\prime}(1)=1$, find $f^{\prime}(\pi/2)$ if $\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}$.

Find the derivative of $h(x)=f(g(x))$ if $$$f(x)=x^2+1$ and $g(x)=3\sqrt{x}$ .

$\bf{Q:}$ Find $(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'$

Practice: trig and inverse trig derivatives

Find the derivative of the following trigonometric and inverse trigonometric functions:

a) $f(x)=\arcsin(3x^2)$

Find the derivative of $f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}$

Find $\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right)$.

Practice: Chain Rule

Find the derivative of $y=\ln(\arctan x)$

The derivative of $y=3^{e^x}$ is

The derivative of $y=3^{e^x}$ is

The derivative of $y=3^{e^x}$ is

The derivative of $y=3^{e^x}$ is

Find $\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right)$.

Find the derivative of the following function.

Find $g^{\prime}(0)$ for $g(x)=\sqrt{f(x^2)}+f(x)$, given $f(0)=2$ and $f^{\prime}(0)=1$.

If $2-g\left(x\right)=x^2+2\left[f\left(x\right)\right]^2-x^3g\left(x\right)$, $f\left(0\right)=1\ ,\ f'\left(0\right)=-2,\ g\left(0\right)=-1\ \text{and}\ \ g''\left(0\right)=3$, then the value of $f''\left(0\right)$ is equal to

Find $g'(0)$ for $g(x)=\sqrt{f(x^2)} + f(x),$ given $f(0)=2 \text{ and }f'(0)=1.$

Find the derivative of $h(x)=f(g(x))$ if $$$f(x)=x^2+1$ and $g(x)=3\sqrt{x}$ .

Find the derivative of $h(x)=f(g(x))$ if $$$f(x)=x^2+1$ and $g(x)=3\sqrt{x}$ .

$\bf{Q:}$ Find $(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'$

If $f(x)=\tan^{-1}\left(3^{\sin x}\right)$, find $f'(\pi)$.

Find the derivative of the following functions at the given values

a) $f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1}$ at $x=0$

b) $g\left(x\right)=\left(\cos x\right)^{e^x}$ at $x=0$

$\bf{Q:}$ Find $(3^{3x-x^{4}}+2^{\cos^{-1}(x)}-2025)'$