The derivative of a product and of a composite function

Calculus, grade 10 (5 units) Lesson 15 of 26Learning

The derivative of a product and of a composite function

The derivative of a product

The derivative of a product of functions is not the product of the derivatives. For example, $\displaystyle {x\cdot x}={x^2}$ , whose derivative is $\displaystyle 2x$ , but $\displaystyle {(x)'\cdot(x)'}={1}$ . The right rule:

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

\displaystyle =

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

The example below differentiates $\displaystyle (x^2+1)(x^3-2x)$ by the rule:

$\displaystyle {f(x)}={x^2+1}$ , $\displaystyle {f'(x)}={2x}$ ; $\displaystyle {g(x)}={x^3-2x}$ , $\displaystyle {g'(x)}={3x^2-2}$ .
The derivative: $\displaystyle 2x(x^3-2x)+(x^2+1)(3x^2-2)$ .
We expand the brackets and collect like terms: $\displaystyle {2x^4-4x^2+3x^4-2x^2+3x^2-2}={5x^4-3x^2-2}$ .

A check: $\displaystyle {(x^2+1)(x^3-2x)}={x^5-x^3-2x}$ , and its derivative is the same $\displaystyle 5x^4-3x^2-2$ .

The proof of the product rule

The proof below writes the difference quotient of $\displaystyle f\cdot g$ , adds and subtracts $\displaystyle f(x)g(x+h)$ , and shows that as $\displaystyle h\to0$ it approaches $\displaystyle f'(x)g(x)+f(x)g'(x)$ .

The difference quotient of the product: $\displaystyle \frac{f(x+h)g(x+h)-f(x)g(x)}{h}$ .
We add and subtract $\displaystyle f(x)g(x+h)$ in the numerator: $\displaystyle \frac{f(x+h)g(x+h)-f(x)g(x+h)+f(x)g(x+h)-f(x)g(x)}{h}$ .
We take out a common factor from each pair: $\displaystyle \frac{f(x+h)-f(x)}{h}\cdot g(x+h)+f(x)\cdot\frac{g(x+h)-g(x)}{h}$ .
As $\displaystyle h\to0$ : $\displaystyle \frac{f(x+h)-f(x)}{h}\to f'(x)$ , $\displaystyle g(x+h)\to g(x)$ , and $\displaystyle \frac{g(x+h)-g(x)}{h}\to g'(x)$ .
So the difference quotient approaches $\displaystyle f'(x)g(x)+f(x)g'(x)$ , and that is the derivative of the product.

A composite function

In $\displaystyle (2x+1)^3$ there are two steps: first $\displaystyle {u}={2x+1}$ , then the result is raised to the power 3. Such a function is a composite function, $\displaystyle f(u(x))$ :

$\displaystyle {u(x)}={2x+1}$ – the inner function, the first step.
$\displaystyle {f(u)}={u^3}$ – the outer function, what is done to the result.

More examples: in $\displaystyle (x^2+1)^4$ the inner function is $\displaystyle x^2+1$ and the outer $\displaystyle u^4$ ; in $\displaystyle |x-3|$ the inner is $\displaystyle x-3$ and the outer $\displaystyle |u|$ .

The derivative of a composite function

If $\displaystyle u$ changes 2 times as fast as $\displaystyle x$ , and $\displaystyle f$ changes 3 times as fast as $\displaystyle u$ , then $\displaystyle f$ changes $\displaystyle {3\cdot2}={6}$ times as fast as $\displaystyle x$ . The rates of change multiply:

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

\displaystyle =

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

$\displaystyle f'(u)$ – the outer derivative: differentiate the outer function, leaving the inner one inside it as it is.
$\displaystyle u'(x)$ – the inner derivative.

The example below differentiates $\displaystyle (x^2+1)^3$ :

The outer function is $\displaystyle u^3$ , and its derivative is $\displaystyle {3u^2}={3(x^2+1)^2}$ .
The inner function is $\displaystyle x^2+1$ , and its derivative is $\displaystyle 2x$ .
$\displaystyle {[(x^2+1)^3]'}={3(x^2+1)^2\cdot2x}={6x(x^2+1)^2}$ .

A linear inner function

When the inner function is linear, $\displaystyle {u}={ax+b}$ , the inner derivative is the number $\displaystyle a$ :

\displaystyle [(ax+b)^n]'

\displaystyle =

\displaystyle n\cdot a(ax+b)^{n-1}

For example, $\displaystyle {[(2x-1)^4]'}={4\cdot2(2x-1)^3}={8(2x-1)^3}$ .

For the same reason, if $\displaystyle {g(x)}={f(ax)}$ then $\displaystyle {g'(x)}={a\cdot f'(ax)}$ . The graph of $\displaystyle g$ is the graph of $\displaystyle f$ squeezed horizontally by $\displaystyle a$ , so all its slopes are $\displaystyle a$ times as large.

Exercise 1

Differentiate by the product rule.
Write the answer without brackets.

/ – 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 by the rule for a composite function.

/ – 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 is $\displaystyle {f(x)}={x(x-2)^3}$ .

Find $\displaystyle f'(x)$ , and factor it.

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

/ – 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

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

Solve the inequality $\displaystyle f'(x)>0$ : where does $\displaystyle f$ rise?

Work out the value of $\displaystyle f$ at the extremum point.

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

/ – 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

Mark the extremum point on the graph.
What does the graph have at $\displaystyle {x}={2}$ ?

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

Exercise 4

The function is $\displaystyle {f(x)}={(2x-3)^4}$ .

Work out the slope of the tangent to the graph at the point where $\displaystyle {x}={2}$ .

$\displaystyle f(x)=(2x-3)^{4}$

/ – 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 at this point on the board.

\displaystyle f(x)=(2x-3)^{4}

Exercise 5

The functions are $\displaystyle {f(x)}={x^2-4x}$ and $\displaystyle {g(x)}={f(3x)}$ .

Find $\displaystyle g'(x)$ .

$\displaystyle f(x)=x^{2}-4x,\qquad g(x)=f(3x)$

/ – 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 $\displaystyle {g'(x)}={0}$ .

$\displaystyle f$ has a minimum at $\displaystyle {x}={2}$ .
Explain by the horizontal squeeze why $\displaystyle g$ has a minimum at $\displaystyle {x}={\frac{2}{3}}$ .

Formulas: calculus, grade 10 (5 units)

Open as a page

Algebra

The square of a sum and of a difference
$\displaystyle (a\pm b)^2$ $\displaystyle =$ $\displaystyle a^2\pm2ab+b^2$
The difference of two squares
$\displaystyle a^2-b^2$ $\displaystyle =$ $\displaystyle (a-b)(a+b)$
The cube of a sum and of a difference
$\displaystyle (a\pm b)^3$ $\displaystyle =$ $\displaystyle a^3\pm3a^2b+3ab^2\pm b^3$
The sum and the difference of two cubes
$\displaystyle a^3\pm b^3$ $\displaystyle =$ $\displaystyle (a\pm b)(a^2\mp ab+b^2)$
The quadratic formula for $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Factoring a trinomial – $\displaystyle x_1$ , $\displaystyle x_2$ the roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle ax^2+bx+c$ $\displaystyle =$ $\displaystyle a(x-x_1)(x-x_2)$

The properties of a function

An even function – the graph is symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – the graph is symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transformations of a graph

A shift of $\displaystyle k$ units up ( $\displaystyle k<0$ – down)
$\displaystyle y=f(x)+k$
A shift of $\displaystyle p$ units right ( $\displaystyle p<0$ – left)
$\displaystyle y=f(x-p)$
A vertical stretch ( $\displaystyle |a|>1$ ) or squeeze ( $\displaystyle |a|<1$ )
$\displaystyle y=a\cdot f(x)$
A horizontal squeeze by $\displaystyle a$ ( $\displaystyle a>1$ ) or stretch ( $\displaystyle 0<a<1$ )
$\displaystyle y=f(ax)$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$

The derivative

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the secant
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
The derivative at a point – the slope of the tangent
$\displaystyle f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}$
The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a constant
$\displaystyle (c)'$ $\displaystyle =$ $\displaystyle 0$
The derivative of a constant multiple
$\displaystyle (c\cdot f(x))'$ $\displaystyle =$ $\displaystyle c\cdot f'(x)$
The derivative of a sum and of a difference
$\displaystyle (f(x)\pm g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\pm g'(x)$
The instantaneous speed – the derivative of the distance with respect to time
$\displaystyle v(t)$ $\displaystyle =$ $\displaystyle s'(t)$

Rules of differentiation

The derivative of a product of functions
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient of functions
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
The derivative of a composite function – the outer derivative times the inner derivative
$\displaystyle [f(u(x))]'$ $\displaystyle =$ $\displaystyle f'(u)\cdot u'(x)$
The derivative of $\displaystyle \frac{1}{x}$
$\displaystyle \left(\frac{1}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{1}{x^2}$
The derivative of $\displaystyle \sqrt{x}$
$\displaystyle (\sqrt{x})'$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x}}$ $\displaystyle x>0$
The derivative of the root of a function
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

The tangent to a graph

The slope of the tangent at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle m=f'(x_1)$
The equation of the tangent
$\displaystyle y-y_1=m(x-x_1)$

Rising, falling and extremum points

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A point that may be an extremum
$\displaystyle f'(x)=0$

Asymptotes

A vertical asymptote – $\displaystyle a$ a zero of the denominator that is not a zero of the numerator
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle b$ the value the function approaches as $\displaystyle x$ grows or falls without end
$\displaystyle y=b$