The graph of a function and the graph of its derivative

Exponential and logarithmic functions, grade 12 (4 units) Lesson 15 of 20Learning

The graph of a function and the graph of its derivative

The sign of the derivative

The derivative $\displaystyle f'(x)$ is the slope of the tangent to the graph of $\displaystyle f$ , so its sign tells where the graph of $\displaystyle f$ goes:

Where $\displaystyle f'(x)>0$ , the function $\displaystyle f$ increases.
Where $\displaystyle f'(x)<0$ , the function $\displaystyle f$ decreases.
At a point where $\displaystyle {f'(x)}={0}$ and the sign of $\displaystyle f'$ changes, $\displaystyle f$ has an extremum point: minus before and plus after – a minimum; plus before and minus after – a maximum.
At a point where $\displaystyle {f'(x)}={0}$ and the sign does not change, there is no extremum point.

So the zeros of the graph of the derivative are the $\displaystyle x$ -coordinates of the extremum points of $\displaystyle f$ , and the intervals where the graph of the derivative is above the $\displaystyle x$ -axis are the intervals where $\displaystyle f$ increases.

A function and its derivative on the same axes

The graph below shows $\displaystyle {f(x)}={xe^x}$ , and dashed, its derivative $\displaystyle {g(x)}={f'(x)}={e^x(x+1)}$ :

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

\displaystyle g(x)=e^{x}(x+1)

$\displaystyle {f'(x)}={0}$ only when $\displaystyle {x}={-1}$ , since $\displaystyle e^x>0$ always.
Left of $\displaystyle -1$ the graph of $\displaystyle f'$ is below the $\displaystyle x$ -axis, and the graph of $\displaystyle f$ decreases; right of it $\displaystyle f'$ is above the axis, and $\displaystyle f$ increases.
So $\displaystyle f$ has a minimum when $\displaystyle {x}={-1}$ : $\displaystyle \left(-1,-\frac{1}{e}\right)$ .
As $\displaystyle x\to-\infty$ both graphs approach $\displaystyle {y}={0}$ : it is the horizontal asymptote of both.

Reading

\displaystyle f

from the graph of its derivative

The graph of $\displaystyle f'$ alone tells how $\displaystyle f$ behaves, even without its formula:

Where $\displaystyle f$ increases and where it decreases – by the side of the $\displaystyle x$ -axis the graph of the derivative is on.
Where $\displaystyle f$ has extremum points, and of which kind – by the zeros of $\displaystyle f'$ and the change of sign.
The height of $\displaystyle f$ itself (for instance $\displaystyle f(0)$ ) cannot be read from the graph of the derivative: that needs the formula of $\displaystyle f$ or a point on its graph.

A parameter from the graph of the derivative

The function $\displaystyle {f(x)}={\frac{e^{x-2}}{x-c}}$ is given, $\displaystyle c$ a parameter. The drawing shows the graph of the derivative $\displaystyle f'(x)$ : it crosses the $\displaystyle x$ -axis where $\displaystyle {x}={2}$ , and it has the vertical asymptote $\displaystyle {x}={1}$ .

\displaystyle g(x)=\frac{e^{x-2}(x-2)}{(x-1)^{2}}

\displaystyle x=1

\displaystyle (2,0)

The example below differentiates $\displaystyle f$ and finds $\displaystyle c$ from the zero of the derivative:

f'(x)=\frac{e^{x-2}(x-c)-e^{x-2}\cdot1}{(x-c)^2}=\frac{e^{x-2}(x-c-1)}{(x-c)^2}

$\displaystyle e^{x-2}>0$ , so $\displaystyle {f'(x)}={0}$ only when $\displaystyle {x}={c+1}$ .
From the drawing $\displaystyle {f'(2)}={0}$ , so $\displaystyle {c+1}={2}$ , that is $\displaystyle {c}={1}$ .
A check: when $\displaystyle {c}={1}$ the denominator is 0 where $\displaystyle {x}={1}$ – the vertical asymptote in the drawing.

Telling the graph of a function from the graph of its derivative

When two graphs are shown together and one is the derivative of the other, check:

At an extremum point of one graph – does the other cross the $\displaystyle x$ -axis there? If so, the other may be the derivative.
Where one graph increases – is the other above the $\displaystyle x$ -axis there?
One condition that fails is enough: then the other graph is not the derivative.

Exercise 1

The function $\displaystyle {f(x)}={(x-1)e^x}$ is given.

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=(x-1)e^{x}$

/ – 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

Find where $\displaystyle f$ decreases: solve an inequality.

The drawing shows $\displaystyle f$ , and dashed, its derivative.
Mark the extremum point of $\displaystyle f$ and the point where the graph of the derivative crosses the $\displaystyle x$ -axis.

\displaystyle f(x)=(x-1)e^{x}

\displaystyle g(x)=\,?

Exercise 2

The drawing shows the graph of the derivative $\displaystyle {f'(x)}={(x+2)(x-1)e^x}$ of a function $\displaystyle f$ .
Mark the points where the graph of the derivative crosses the $\displaystyle x$ -axis.

\displaystyle g(x)=(x+2)(x-1)e^{x}

Where does $\displaystyle f$ increase, and where does it decrease?

What are the $\displaystyle x$ -coordinates of the extremum points of $\displaystyle f$ ? State their kind.

Exercise 3

The drawing shows the graphs of two functions, $\displaystyle p$ and $\displaystyle q$ .
One of them is $\displaystyle {f(x)}={\ln(x^2+1)}$ , and the other is its derivative.
Which is $\displaystyle f$ , and which is $\displaystyle f'$ ? Explain from the graphs.

\displaystyle p(x)=\,?

\displaystyle q(x)=\,?

Differentiate $\displaystyle f$ , and check the answer.

/ – 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 4

The function $\displaystyle {f(x)}={\frac{e^x}{x-c}}$ is given, $\displaystyle c$ a parameter.
The drawing shows the graph of the derivative $\displaystyle f'(x)$ .
The graph of the derivative crosses the $\displaystyle x$ -axis where $\displaystyle {x}={3}$ .

\displaystyle g(x)=\,?

\displaystyle x=2

Differentiate $\displaystyle f$ , and calculate $\displaystyle c$ .

Put in the $\displaystyle c$ you found.
What is the domain of $\displaystyle f$ ? Where does it increase, and where does it decrease?

Calculate $\displaystyle f(3)$ .
Write the extremum point of $\displaystyle f$ , and state its kind.

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

/ – 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

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
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$