Derivative of Exponential Functions


The functions $e^{-x}$, $e^{3x+2}$ and $e^{x^2+2x-1}$ are all of the form $e^{f(x)}$.
$e^{f(x)} \gt 0$ for all $x$, no matter what the function $f(x)$.
$$\displaystyle \dfrac{d}{dx}e^x = e^x$$
$$\displaystyle \dfrac{d}{dx}e^{f(x)} = e^{f(x)} \times f'(x)$$

Example 1

Find $\displaystyle \dfrac{dy}{dx}$ if $y=e^{4x}$.

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

Example 2

Find $\displaystyle \dfrac{dy}{dx}$ if $y=x^2e^{3x}$.

\( \begin{align} \displaystyle \require{color}
\dfrac{dy}{dx} &= \dfrac{d}{dx}x^2 \times e^{3x} + x^2 \times \dfrac{d}{dx}e^{3x} &\color{red} \text{product rule}\\
&= 2x \times e^{3x} + x^2 \times e^{3x} \times \dfrac{d}{dx}3x \\
&= 2xe^{3x} + x^2 \times e^{3x} \times 3 \\
&= 2xe^{3x} + 3x^2 e^{3x} \\
\end{align} \)

Example 3

Find $\displaystyle \dfrac{dy}{dx}$ if $y=\dfrac{e^{2x}}{x}$.

\( \begin{align} \displaystyle \require{color}
\dfrac{dy}{dx} &= \dfrac{\dfrac{d}{dx}e^{2x} \times x – e^{2x} \times \dfrac{d}{dx}x}{x^2} &\color{red} \text{quotient rule} \\
&= \dfrac{2e^{2x} \times x – e^{2x} \times 1}{x^2} \\
&= \dfrac{2xe^{2x} – e^{2x}}{x^2} \\
\end{align} \)

Extension Examples


These Extension Examples require to have some prerequisite skills including;
\( \begin{align} \displaystyle
\dfrac{d}{dx}\sin{x} &= \cos{x} \\
\dfrac{d}{dx}\cos{x} &= -\sin{x} \\
\dfrac{d}{dx}\log_e{x} &= \dfrac{1}{x} \\
\end{align} \)

Example 4

Find $\displaystyle \dfrac{dy}{dx}$ of $y=e^x\sin{x}$, known that $\dfrac{d}{dx}\sin{x} = \cos{x}$.

\( \begin{align} \displaystyle
\dfrac{dy}{dx} &= \dfrac{d}{dx}e^x \times \sin{x} + e^x \times \dfrac{d}{dx}\sin{x} \\
&= e^x \times \sin{x} + e^x \times \cos{x} \\
&= e^x\sin{x} + e^x\cos{x} \\
\end{align} \)

Example 5

Find $\displaystyle \dfrac{dy}{dx}$ of $y=e^{\cos{x}}$, known that $\dfrac{d}{dx}\cos{x} = -\sin{x}$.

\( \begin{align} \displaystyle
\dfrac{dy}{dx} &= e^{\cos{x}} \times \dfrac{d}{dx}\cos{x} \\
&= e^{\cos{x}} \times -\sin{x} \\
&= -e^{\cos{x}}\sin{x} \\
\end{align} \)

Example 6

Find $\displaystyle \dfrac{dy}{dx}$ of $y=e^{2x}\log_e{x^2}$, known that $\dfrac{d}{dx}\log_e{x} = \dfrac{1}{x}$.

\( \begin{align} \displaystyle
\dfrac{dy}{dx} &= \dfrac{d}{dx}e^{2x} \times \log_e{x^2} + e^{2x} \times \dfrac{d}{dx}\log_e{x^2} \\
&= 2e^{2x} \times \log_e{x^2} + e^{2x} \times \dfrac{1}{x^2} \times \dfrac{d}{dx}x^2 \\
&= 2e^{2x}\log_e{x^2} + e^{2x} \times \dfrac{1}{x^2} \times 2x \\
&= 2e^{2x}\log_e{x^2} + \dfrac{2e^{2x}}{x} \\
\end{align} \)


Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Divisibility Proof Double-Angle Formula Equation Exponent Exponential Function Factorials 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 Proof Pythagoras Theorem Quadratic Quadratic Factorise Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume




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