# Higher order derivatives

### Higher order derivatives

#### Lessons

Note
If $f'(x)$ is the derivative of $f(x)$, then we say that $f"(x)$ is the $2^{nd}$ derivative of $f(x)$. Similarly, $f^{(n)}(x)$ is the $n'th$ derivative of $f(x)$.
• Introduction
The concept of higher order derivatives

• 1.
$1^{st}$ and $2^{nd}$ derivatives.
Find the first and second derivative for the following functions:
a)
$f(x)=x^4+5x^2+3x+2$

b)
$f(t)=\sin (2t)$

c)
$g(s)=(2s+5s^2)^7$

d)
$y=5$

e)
$f(x)=5 \ln x$

• 2.
$2^{nd}$ derivatives with implicit differentation
Find $y"$ by implicit differentiation for the following functions:
a)
$x^2+y^2=9$

b)
$x^2+xy=9$

• 3.
Derivatives with repeating patterns
Find $f^{(100)}(x)$ for the following functions:
a)
$f(x)= \sin (x)$

b)
$f(x)=e^{(2x)}$