Higher Derivatives

Given a function $f(x)$, the derivative $f'(x)$ is known as the first derivative.
The second derivative of $f(x)$ is the derivative of $f'(x)$, which is $f^{\prime \prime}(x)$ or the derivative of the first derivative.
$$ \displaystyle \begin{align}
f'(x) &= \dfrac{d}{dx}f(x) \\
f^{\prime \prime}(x) &= \dfrac{d}{dx}f'(x) \\
f^{(3)}(x) &= \dfrac{d}{dx}f^{\prime \prime}(x) \\
f^{(4)}(x) &= \dfrac{d}{dx}f^{(3)}(x) \\
&\cdots \\
f^{(n)}(x) &= \dfrac{d}{dx}f^{(n-1)}(x) \\
\end{align} $$
We can continue to differentiate to obtain higher derivatives.
The $n$th derivative if $y$ with respect to $x$ is obtained by differentiating $y=f(x)$ $n$ times. We use the notation $f^{(n)}(x)$ or $\dfrac{d^ny}{dx^n}$ for the $n$th derivative.

Let’s do some practices for this now!

Example 1

Find $f^{\prime \prime}(x)$ given that $f(x)=x^4-3x^2-4x+5$.

Example 2

Find $f^{\prime \prime}(x)$ given that $f(x)=(x^2-1)^5$.

Extension Examples

These Extension Examples require to have some prerequisite skills including;
\( \begin{align} \displaystyle
\dfrac{d}{dx}\sin{x} &= \cos{x} \\
\dfrac{d}{dx}\cos{x} &= -\sin{x} \\
\dfrac{d}{dx}e^x &= e^x \\
\end{align} \)

Example 3

Find $f^{(3)}(x)$ if $f(x)=\sin{(2x)}$, given that $(\sin{x})’=\cos{x}$ and $(\cos{x})’=-\sin{x}$.

Example 4

Find $\dfrac{d^{(3)}y}{dx^{(3)}}$ if $y=e^{2x}$, given that $\dfrac{d}{dx}e^x = e^x$.

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