The number e and derivatives

Exponential and logarithmic functions, grade 12 (4 units) Lesson 12 of 20Learning

The number e and derivatives

Reminder: the rules of differentiation

The rules of differentiation from grades 10 and 11 hold for the new functions too:

$\displaystyle {(x^n)'}={nx^{n-1}}$ , $\displaystyle {(\sqrt{x})'}={\frac{1}{2\sqrt{x}}}$ , and the derivative of a sum is the sum of the derivatives.
The derivative of a product and the derivative of a quotient:

\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}}

The derivative of a composite function: the outer derivative times the inner derivative.

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

\displaystyle =

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

The equation of the tangent to the graph at the point where $\displaystyle {x}={a}$ :

\displaystyle y-f(a)=f'(a)(x-a)

The number

\displaystyle e

The number $\displaystyle e$ is a constant, like $\displaystyle \pi$ : $\displaystyle e\approx2.718$ . One way to reach it is the expression $\displaystyle \left(1+\frac{1}{n}\right)^n$ as $\displaystyle n$ grows:

$\displaystyle {n}={10}$ : $\displaystyle \left(1+\frac{1}{10}\right)^{10}\approx2.594$
$\displaystyle {n}={100}$ : $\displaystyle \approx2.705$
$\displaystyle {n}={1000}$ : $\displaystyle \approx2.717$
$\displaystyle {n}={1{,}000{,}000}$ : $\displaystyle \approx2.71828$

The function $\displaystyle e^x$ is the exponential function whose base is $\displaystyle e$ . Its graph runs between the graphs of $\displaystyle 2^x$ and of $\displaystyle 3^x$ :

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

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

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

\displaystyle y=x+1

Like every exponential function, $\displaystyle e^x$ is positive and increasing, passes through $\displaystyle (0,1)$ , and its horizontal asymptote is $\displaystyle {y}={0}$ .
The tangent to the graph of $\displaystyle e^x$ at the point $\displaystyle (0,1)$ is $\displaystyle {y}={x+1}$ : its slope is 1. That is what is special about the base $\displaystyle e$ .

The natural logarithm

\displaystyle \ln x

The logarithm to the base $\displaystyle e$ is called the natural logarithm, written $\displaystyle \ln$ :

\displaystyle \ln x=\log_{e}x

$\displaystyle \ln x$ is defined only when $\displaystyle x>0$ .
$\displaystyle {\ln1}={0}$ , $\displaystyle {\ln e}={1}$ , $\displaystyle {\ln e^x}={x}$ , $\displaystyle {e^{\ln x}}={x}$ .
The laws of logarithms hold for $\displaystyle \ln$ too: $\displaystyle {\ln(xy)}={\ln x+\ln y}$ , $\displaystyle {\ln\frac{x}{y}}={\ln x-\ln y}$ , $\displaystyle {\ln x^b}={b\ln x}$ .
$\displaystyle {e^x}={5}$ means $\displaystyle {x}={\ln5}$ .

The derivatives of

\displaystyle e^x

and of

\displaystyle \ln x

The derivative of $\displaystyle e^x$ is $\displaystyle e^x$ itself: at every point the slope of the tangent equals the height of the graph. The derivative of $\displaystyle \ln x$ is $\displaystyle \frac{1}{x}$ , in the domain $\displaystyle x>0$ .

\displaystyle (e^{x})'

\displaystyle =

\displaystyle e^{x}

\displaystyle (\ln x)'

\displaystyle =

\displaystyle \frac{1}{x}

The example below differentiates a sum of the two new functions and a polynomial:

\left(3e^x-2\ln x+x^2\right)'=3e^x-\frac{2}{x}+2x

The derivatives of

\displaystyle e^{f(x)}

and of

\displaystyle \ln f(x)

When $\displaystyle f(x)$ is a polynomial, $\displaystyle e^{f(x)}$ and $\displaystyle \ln f(x)$ are composite functions: the outer derivative times the inner derivative $\displaystyle f'(x)$ .

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

\displaystyle =

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

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

\displaystyle =

\displaystyle \frac{f'(x)}{f(x)}

$\displaystyle {\left(e^{x^2-3x}\right)'}={(2x-3)e^{x^2-3x}}$ .
$\displaystyle {\left(\ln(x^2+1)\right)'}={\frac{2x}{x^2+1}}$ .
$\displaystyle \ln f(x)$ is defined only where $\displaystyle f(x)>0$ : the domain of $\displaystyle \ln(x^2+1)$ is every $\displaystyle x$ , that of $\displaystyle \ln(x-2)$ only $\displaystyle x>2$ .

A product and a quotient with a polynomial

The examples below differentiate a product and a quotient, and take out a common factor, so that $\displaystyle {f'(x)}={0}$ is easy to solve later:

\left(xe^{2x}\right)'=1\cdot e^{2x}+x\cdot2e^{2x}=e^{2x}(1+2x)

\left(\frac{\ln x}{x}\right)'=\frac{\frac{1}{x}\cdot x-\ln x\cdot1}{x^2}=\frac{1-\ln x}{x^2}

The equation of a tangent

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

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

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 if there is one.

/ – 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 $\displaystyle {f(x)}={xe^{-x}}$ is given.

Find $\displaystyle f'(x)$ .

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

/ – 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

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

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

/ – 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 $\displaystyle (0,0)$ .

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

\displaystyle (0,0)

Exercise 4

The function $\displaystyle {f(x)}={\ln(x^2-3)}$ is given.

Find the domain of the function.

Calculate $\displaystyle f'(2)$ .

$\displaystyle f(x)=\ln(x^{2}-3)$

/ – 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

$\displaystyle {f(2)}={\ln1}={0}$ .
Write the equation of the tangent to the graph at the point $\displaystyle (2,0)$ .

\displaystyle f(x)=\ln(x^{2}-3)

\displaystyle (2,0)

Exercise 5

The function $\displaystyle {f(x)}={e^{2x}}$ is given.
At one point of the graph the slope of the tangent is 2.
Find the point: solve an equation, and write the point’s coordinates.

Formulas: exponential and logarithmic functions, grade 12 (4 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
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

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 $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
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 and the area

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int e^{mx+b}\,dx=\frac{e^{mx+b}}{m}+C$ $\displaystyle m\neq0$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle \int\frac{1}{mx+b}\,dx=\frac{\ln|mx+b|}{m}+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$