Higher Derivatives

Higher Derivatives

Given a function $f(x)$, the derivative $f^{\prime}(x)$ is known as the first derivative.
The second derivative of $f(x)$ is the derivative of $f^{\prime}(x)$, which is $f^{\prime \prime}(x)$ or the derivative of the first derivative.
$$ \displaystyle \begin{align}
f^{\prime}(x) &= \dfrac{d}{dx}f(x) \\
f^{\prime \prime}(x) &= \dfrac{d}{dx}f'(x) \\
f^{(3)}(x) &= \dfrac{d}{dx}f^{\prime \prime}(x) \\
f^{(4)}(x) &= \dfrac{d}{dx}f^{(3)}(x) \\
&\cdots \\
f^{(n)}(x) &= \dfrac{d}{dx}f^{(n-1)}(x) \\
\end{align} $$
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 practice for this now!

Example 1

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

\( \begin{align} \displaystyle
f^{\prime}(x) &= (x^4-3x^2-4x+5)’ \\
&= 4x^3-6x-4 \\
f^{\prime \prime}(x) &= (4x^3-6x-4)’ \\
&= 12x^2-6
\end{align} \)

Example 2

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

\( \begin{align} \displaystyle
f^{\prime}(x) &= \left[(x^2-1)^5\right]’ \\
&= 5(x^2-1)^{5-1} \times (x^2-1)’ \\
&= 5(x^2-1)^{4} \times 2x \\
&= 10x(x^2-1)^{4} \\
f^{\prime \prime}(x) &= \left[10x(x^2-1)^{4}\right]’ \\
&= 40x(x^2-1)^{4-1} \times (x^2-1)’ \\
&= 40x(x^2-1)^{3} \times 2x \\
&= 80x^2(x^2-1)^{3}
\end{align} \)

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 $(\sin{x})^{\prime}=\cos{x}$ and $(\cos{x})^{\prime}=-\sin{x}$.

\( \begin{align} \displaystyle
f^{\prime}(x) &= \left[\sin{(2x)}\right]’ \\
&= 2\cos{(2x)} \\
f^{\prime \prime}(x) &= \left[2\cos{(2x)})\right]’ \\
&= -4\sin{(2x)} \\
f^{(3)} &= \left[-4\sin{(2x)}\right]’ \\
&= -8\cos{(2x)}
\end{align} \)

Example 4

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

\( \begin{align} \displaystyle
\dfrac{dy}{dx} &= e^{2x} \times \dfrac{d}{dx}2x \\
&= e^{2x} \times 2 \\
&= 2e^{2x} \\
\dfrac{d^{2}y}{dx^{2}} &= 2e^{2x} \times \dfrac{d}{dx}2x \\
&= 2e^{2x} \times 2 \\
&= 4e^{2x} \\
\dfrac{d^{3}y}{dx^{3}} &= 4e^{2x} \times \dfrac{d}{dx}2x \\
&= 4e^{2x} \times 2 \\
&= 8e^{2x}
\end{align} \)

Unlock your full learning potential—download our expertly crafted slide files for free and transform your self-study sessions!

Discover more enlightening videos by visiting our YouTube channel!

 

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume




Related Articles

Responses

Your email address will not be published. Required fields are marked *