The derivative of an exponential function

Exponential and logarithmic functions, grade 12 (5 units) Lesson 12 of 23Learning

The derivative of an exponential function

Reminder from grade 11: the rules of differentiation

The rules of differentiation you learnt in grade 11 hold for the new functions too:

The derivative of a sum is the sum of the derivatives, and $\displaystyle {(c\cdot f(x))'}={c\cdot f'(x)}$ .
The derivative of a product, of a quotient and of a composite function:

\displaystyle \left[f(x)\cdot g(x)\right]'

\displaystyle =

\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)

\displaystyle \left[\frac{f(x)}{g(x)}\right]'

\displaystyle =

\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{\left[g(x)\right]^{2}}

\displaystyle \left[f(u(x))\right]'

\displaystyle =

\displaystyle f'(u)\cdot u'(x)

The second derivative $\displaystyle f''(x)$ is the derivative of $\displaystyle f'(x)$ . Where $\displaystyle f''(x)>0$ the graph is concave up, and where $\displaystyle f''(x)<0$ it is concave down.
An inflection point is a point of the graph where the concavity changes: $\displaystyle f''$ changes sign there.

The slope of

\displaystyle a^x

at the point

\displaystyle (0,1)

All the graphs of $\displaystyle {y}={a^x}$ pass through $\displaystyle (0,1)$ , but at different slopes. We work out the slope roughly, as the slope of the chord between $\displaystyle {x}={0}$ and $\displaystyle {x}={0.001}$ , that is $\displaystyle \frac{a^{0.001}-1}{0.001}$ :

$\displaystyle {a}={2}$ : $\displaystyle \approx0.693$ .
$\displaystyle {a}={2.7}$ : $\displaystyle \approx0.994$ .
$\displaystyle {a}={3}$ : $\displaystyle \approx1.099$ .

There is one base for which the slope at the point $\displaystyle (0,1)$ is exactly 1 – the base $\displaystyle e\approx2.718$ . The tangent to the graph of $\displaystyle e^x$ at the point $\displaystyle (0,1)$ is $\displaystyle {y}={x+1}$ :

\displaystyle f(x)=2^{x}

\displaystyle g(x)=e^{x}

\displaystyle h(x)=3^{x}

\displaystyle y=x+1

The derivative of

\displaystyle e^x

The slope of $\displaystyle e^x$ at every point follows from its slope at the point $\displaystyle (0,1)$ :

The slope of the chord between $\displaystyle x$ and $\displaystyle x+h$ is $\displaystyle {\frac{e^{x+h}-e^x}{h}}={e^x\cdot\frac{e^h-1}{h}}$ .
As $\displaystyle h$ comes close to 0, $\displaystyle \frac{e^h-1}{h}$ comes close to the slope at the point $\displaystyle (0,1)$ , that is to 1.
So the slope at the point where $\displaystyle x$ is is $\displaystyle e^x$ : at every point the slope of the tangent equals the height of the graph.

\displaystyle (e^{x})'

\displaystyle =

\displaystyle e^{x}

The derivative of

\displaystyle e^{f(x)}

$\displaystyle e^{f(x)}$ is a composite function: the outer derivative $\displaystyle e^{f(x)}$ times the inner derivative $\displaystyle f'(x)$ .

\displaystyle \left[e^{f(x)}\right]'

\displaystyle =

\displaystyle e^{f(x)}\cdot f'(x)

$\displaystyle {\left(e^{3x}\right)'}={3e^{3x}}$ .
$\displaystyle {\left(e^{x^2-2x}\right)'}={(2x-2)e^{x^2-2x}}$ .
$\displaystyle {\left(e^{-x}\right)'}={-e^{-x}}$ .

The derivative of

\displaystyle a^x

Every exponential function can be written to the base $\displaystyle e$ : $\displaystyle {a^x}={e^{x\ln a}}$ . By the derivative of $\displaystyle e^{f(x)}$ , with $\displaystyle {f(x)}={x\ln a}$ :

\displaystyle (a^{x})'

\displaystyle =

\displaystyle a^{x}\cdot\ln a

$\displaystyle {(2^x)'}={2^x\ln2}$ , and $\displaystyle {(10^x)'}={10^x\ln10}$ .
The derivative of $\displaystyle a^x$ is $\displaystyle a^x$ times a constant, $\displaystyle \ln a$ .

\displaystyle f(x)=2^{x}

\displaystyle g(x)=2^{x}\ln2

\displaystyle h(x)=3^{x}

\displaystyle k(x)=3^{x}\ln3

In the drawing: $\displaystyle 2^x$ and $\displaystyle 3^x$ , and beside them their derivatives (dashed).

$\displaystyle \ln2\approx0.69<1$ : the graph of $\displaystyle (2^x)'$ is below the graph of $\displaystyle 2^x$ .
$\displaystyle \ln3\approx1.10>1$ : the graph of $\displaystyle (3^x)'$ is above the graph of $\displaystyle 3^x$ .
For the base $\displaystyle e$ , $\displaystyle {\ln e}={1}$ , and the graph of the derivative is the graph of the function itself.

The equation of a tangent

The example below finds the equation of the tangent to the graph of $\displaystyle {f(x)}={e^{2x-2}}$ at the point where $\displaystyle {x}={1}$ :

The point of contact: $\displaystyle {f(1)}={e^0}={1}$ , that is $\displaystyle (1,1)$ .
The slope: $\displaystyle {f'(x)}={2e^{2x-2}}$ , so $\displaystyle {f'(1)}={2}$ .
The tangent: $\displaystyle {y-1}={2(x-1)}$ , that is $\displaystyle {y}={2x-1}$ .

The second derivative and concavity

The example below finds the intervals of concavity and the inflection points of $\displaystyle {f(x)}={e^{-x^2}}$ :

$\displaystyle {f'(x)}={-2xe^{-x^2}}$ .
$\displaystyle {f''(x)}={-2e^{-x^2}+(-2x)(-2x)e^{-x^2}}={(4x^2-2)e^{-x^2}}$ .
$\displaystyle e^{-x^2}>0$ , so the sign of $\displaystyle f''$ is the sign of $\displaystyle 4x^2-2$ : $\displaystyle {f''(x)}={0}$ where $\displaystyle {x}={\pm\frac{\sqrt{2}}{2}\approx\pm0.707}$ .
Concave up where $\displaystyle x<-\frac{\sqrt{2}}{2}$ or $\displaystyle x>\frac{\sqrt{2}}{2}$ , and concave down where $\displaystyle -\frac{\sqrt{2}}{2}<x<\frac{\sqrt{2}}{2}$ .

\displaystyle f(x)=e^{-x^{2}}

\displaystyle P=(-0.707,0.607)

\displaystyle Q=(0.707,0.607)

The inflection points: $\displaystyle \left(\pm\frac{\sqrt{2}}{2},e^{-\frac{1}{2}}\right)\approx(\pm0.707,0.607)$ .
This graph is called a “bell”: concave down round the maximum $\displaystyle (0,1)$ , and concave up on both sides, as it comes close to the asymptote $\displaystyle {y}={0}$ .

Exercise 1

Differentiate.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

Differentiate, and take out a common factor.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

The function $\displaystyle {f(x)}={xe^{2x}}$ is given.
$\displaystyle {f'(x)}={e^{2x}(1+2x)}$ .

Find the second derivative $\displaystyle f''(x)$ , and take out a common factor.

$\displaystyle f(x)=xe^{2x}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Find where the graph is concave up: solve the inequality $\displaystyle f''(x)>0$ .

Mark the inflection point on the graph.

\displaystyle f(x)=xe^{2x}

Exercise 4

The function $\displaystyle {f(x)}={3^x}$ is given.

Calculate the slope of the tangent to the graph at the point where $\displaystyle {x}={0}$ .

$\displaystyle f(x)=3^{x}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Write the equation of the tangent to the graph at the point $\displaystyle (0,1)$ .

\displaystyle f(x)=3^{x}

\displaystyle (0,1)

Exercise 5

The function $\displaystyle {f(x)}={e^{2x}-4x}$ is given.

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=e^{2x}-4x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f'(x)}={0}$ .

Mark the extremum point on the graph.

\displaystyle f(x)=e^{2x}-4x

Formulas: exponential and logarithmic functions, grade 12 (5 units)

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Powers

A product of powers with the same base
$\displaystyle a^{x}\cdot a^{y}$ $\displaystyle =$ $\displaystyle a^{x+y}$
A quotient of powers with the same base
$\displaystyle \frac{a^{x}}{a^{y}}$ $\displaystyle =$ $\displaystyle a^{x-y}$
A power of a power
$\displaystyle (a^{x})^{y}$ $\displaystyle =$ $\displaystyle a^{x\cdot y}$
A power of a product
$\displaystyle (a\cdot b)^{x}$ $\displaystyle =$ $\displaystyle a^{x}\cdot b^{x}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{x}$ $\displaystyle =$ $\displaystyle \frac{a^{x}}{b^{x}}$
A negative exponent
$\displaystyle a^{-x}$ $\displaystyle =$ $\displaystyle \frac{1}{a^{x}}$
The exponent 0
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A rational exponent
$\displaystyle a^{\frac{m}{n}}$ $\displaystyle =$ $\displaystyle \sqrt[n]{a^{m}}$ $\displaystyle a>0$

Logarithms (within the domain)

The definition of the logarithm
$\displaystyle \log_{a}x=b\iff a^{b}=x$ $\displaystyle a>0,\ a\neq1,\ x>0$
$\displaystyle \log_{a}(a^{b})$ $\displaystyle =$ $\displaystyle b$
$\displaystyle a^{\log_{a}x}$ $\displaystyle =$ $\displaystyle x$
The logarithm of a product
$\displaystyle \log_{a}x+\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}(x\cdot y)$
The logarithm of a quotient
$\displaystyle \log_{a}x-\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}\left(\frac{x}{y}\right)$
The logarithm of a power
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
Change of base
$\displaystyle \log_{c}x$ $\displaystyle =$ $\displaystyle \frac{\log_{a}x}{\log_{a}c}$
$\displaystyle \log_{a}b$ $\displaystyle =$ $\displaystyle \frac{1}{\log_{b}a}$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

The exponential and the logarithmic function

Every exponential as a power of $\displaystyle e$
$\displaystyle a^{x}$ $\displaystyle =$ $\displaystyle e^{x\ln a}$
$\displaystyle a^x$ and $\displaystyle \log_a x$ – inverse functions
$\displaystyle y=a^{x}\iff x=\log_{a}y$ $\displaystyle y>0$
The number $\displaystyle e$
$\displaystyle e\approx2.718$

Growth and decay

The amount after $\displaystyle t$ units of time
$\displaystyle f(t)=f(0)\cdot q^{t}$
The growth / decay factor – $\displaystyle p$ the percent per unit of time
$\displaystyle q=1\pm\frac{p}{100}$
The half-life $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$
The doubling time $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle 2$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (a^{x})^{\prime}$ $\displaystyle =$ $\displaystyle a^{x}\cdot\ln a$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
$\displaystyle (\log_{a}x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x\cdot\ln a}$
The derivative of $\displaystyle e^{f(x)}$
$\displaystyle \left[e^{f(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle e^{f(x)}\cdot f^{\prime}(x)$
The derivative of $\displaystyle \ln(f(x))$
$\displaystyle \left[\ln(f(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)}{f(x)}$
The derivative of a product
$\displaystyle \left[f(x)\cdot g(x)\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)\cdot g(x)-f(x)\cdot g^{\prime}(x)}{\left[g(x)\right]^{2}}$
The derivative of a composite function
$\displaystyle \left[f(u(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(u)\cdot u^{\prime}(x)$
The tangent’s equation at the point where $\displaystyle {x}={a}$
$\displaystyle y-f(a)=f^{\prime}(a)(x-a)$

Integrals, the area and accumulation

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int a^{x}\,dx=\frac{a^{x}}{\ln a}+C$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(u(x))\cdot u^{\prime}(x)\,dx=F(u(x))+C$
Where $\displaystyle f(x)\neq0$
$\displaystyle \int\frac{f^{\prime}(x)}{f(x)}\,dx=\ln|f(x)|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$
The accumulation function’s derivative
$\displaystyle \left(\int_{a}^{x}f(t)\,dt\right)^{\prime}=f(x)$