The derivative of a product and of a composite function

Algebra and calculus, grade 10 (4 units) Lesson 11 of 20Learning

The derivative of a product and of a composite function

The derivative of a product

The derivative of a product is not the product of the derivatives. It is a sum of two terms: in each, one factor is differentiated and the other stays as it is.

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

\displaystyle =

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

$\displaystyle f'(x)\cdot g(x)$ – the first factor differentiated, the second kept.
$\displaystyle f(x)\cdot g'(x)$ – the first kept, the second differentiated.

A check by expanding

The example below differentiates $\displaystyle (x^2+1)(x-3)$ in two ways and shows that both give the same derivative.

By expanding the brackets:

(x^2+1)(x-3)=x^3-3x^2+x-3

(x^3-3x^2+x-3)'=3x^2-6x+1

By the product rule, with $\displaystyle {f(x)}={x^2+1}$ and $\displaystyle {g(x)}={x-3}$ :

2x\cdot(x-3)+(x^2+1)\cdot1=2x^2-6x+x^2+1=3x^2-6x+1

Both ways give $\displaystyle 3x^2-6x+1$ .
The product of the derivatives, $\displaystyle {2x\cdot1}={2x}$ , is a different answer – so it is wrong.

A composite function

The function $\displaystyle (2x-1)^3$ works in two steps: first $\displaystyle 2x-1$ is worked out, then the result is raised to the power 3. Such a function is called a composite function.

The inner function – what is worked out first: $\displaystyle 2x-1$ .
The outer function – what is done to the result: the power $\displaystyle u^3$ .
More examples: in $\displaystyle (x^2+1)^4$ the inner function is $\displaystyle x^2+1$ and the outer one $\displaystyle u^4$ ; in $\displaystyle 8(2x-1)^3$ the outer one is $\displaystyle 8u^3$ .

The derivative of a composite function

Differentiate the outer function, keeping the inner one inside it as it is, and multiply the result by the derivative of the inner function.

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

\displaystyle =

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

Why a product: the inner function changes at the rate $\displaystyle g'$ with respect to $\displaystyle x$ , and the outer one at the rate $\displaystyle f'$ with respect to the inner one. A small change in $\displaystyle x$ passes through both rates, so they multiply.

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

\left((2x-1)^3\right)'=3(2x-1)^2\cdot2=6(2x-1)^2

$\displaystyle 3(2x-1)^2$ – the derivative of the outer $\displaystyle u^3$ , with the inner function inside it as it is.
$\displaystyle \cdot2$ – the derivative of the inner $\displaystyle 2x-1$ .

A solved example

The example below differentiates $\displaystyle (x^2+1)^4$ :

\left((x^2+1)^4\right)'=4(x^2+1)^3\cdot2x=8x(x^2+1)^3

The derivative of the outer function: $\displaystyle 4(x^2+1)^3$ .
The derivative of the inner function: $\displaystyle 2x$ .
The brackets stay closed: in factored form it is easy to see where the derivative is 0.

Exercise 1

Differentiate by the product rule, and write the derivative 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 the composite functions.

/ – 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 $\displaystyle {f(x)}={(x^2-4)^2}$ .

Find $\displaystyle f'(x)$ .

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

/ – 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

Work out $\displaystyle f'(1)$ .

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

/ – 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

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

Exercise 4

The function is $\displaystyle {f(x)}={8(2x-1)^3}$ , defined for every $\displaystyle x$ .

Mark the points where the graph of the function meets the axes.

\displaystyle f(x)=8(2x-1)^{3}

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=8(2x-1)^{3}$

/ – 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

Work out $\displaystyle f'\left(\frac{1}{2}\right)$ .

$\displaystyle f(x)=8(2x-1)^{3}$

/ – 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

What is the slope of the tangent to the graph at the point where it meets the $\displaystyle x$ -axis?
Describe that tangent.

/ – 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 5

The function $\displaystyle x(x-1)^3$ is a product, and one of its factors is a composite function.
Differentiate it, and write the derivative as a product of factors.

/ – 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: algebra and calculus, grade 10 (4 units)

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Algebraic technique

The square of a sum
$\displaystyle (a+b)^2$ $\displaystyle =$ $\displaystyle a^2+2ab+b^2$
The square of a difference
$\displaystyle (a-b)^2$ $\displaystyle =$ $\displaystyle a^2-2ab+b^2$
The difference of squares
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^2-b^2$
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 roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
The product of roots
$\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle =$ $\displaystyle \sqrt{ab}$ $\displaystyle a\geq0,\ b\geq0$
The quotient of roots
$\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle =$ $\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle a\geq0,\ b>0$

The properties of a function

An even function – its graph symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – its graph symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The $\displaystyle x$ -coordinate of the vertex of $\displaystyle {y}={ax^2+bx+c}$
$\displaystyle x=\frac{-b}{2a}$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transforming a graph

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

The derivative and the rules of differentiation

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the chord
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
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 derivative of a product
$\displaystyle (f(x)\cdot g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a composite function
$\displaystyle (f(g(x)))'$ $\displaystyle =$ $\displaystyle f'(g(x))\cdot g'(x)$

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
$\displaystyle y-y_1=m(x-x_1)$

The derivative of a square root function

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
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

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
$\displaystyle f'(x)=0$