The derivative function and the rules of differentiation

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

The derivative function and the rules of differentiation

The derivative function

For $\displaystyle {f(x)}={x^2}$ the last lesson found $\displaystyle {f'(1)}={2}$ , $\displaystyle {f'(2)}={4}$ , $\displaystyle {f'(3)}={6}$ : at every point the derivative is twice the $\displaystyle x$ . It can be worked out for every $\displaystyle x$ at once:

\frac{(x+h)^2-x^2}{h}=\frac{2xh+h^2}{h}=2x+h

As $\displaystyle h$ gets closer to 0, $\displaystyle 2x+h$ gets closer to $\displaystyle 2x$ .
So $\displaystyle {f'(x)}={2x}$ : a function that gives, for every $\displaystyle x$ , the slope of the tangent to the graph at its point.
$\displaystyle f'(x)$ is called the derivative function of $\displaystyle f$ , and finding it is called differentiating $\displaystyle f$ .

$\displaystyle x$	$\displaystyle -2$	$\displaystyle -1$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 2$	$\displaystyle 3$
$\displaystyle {f'(x)}={2x}$	$\displaystyle -4$	$\displaystyle -2$	$\displaystyle 0$	$\displaystyle 2$	$\displaystyle 4$	$\displaystyle 6$

The derivative of a power and of a constant

Every power of $\displaystyle x$ is differentiated the way $\displaystyle x^2$ was: the index comes down in front of the $\displaystyle x$ , and the power goes down by 1.

\displaystyle (x^n)'

\displaystyle =

\displaystyle nx^{n-1}

\displaystyle (c)'

\displaystyle =

\displaystyle 0

$\displaystyle {(x^5)'}={5x^4}$ , $\displaystyle {(x^3)'}={3x^2}$ .
$\displaystyle {(x)'}={1}$ : the graph of $\displaystyle {y}={x}$ is a line with slope 1.
The graph of a constant, $\displaystyle {y}={7}$ , is a level line: its slope is 0, so $\displaystyle {(7)'}={0}$ .

A constant multiple and a sum

A number multiplying a function stays in front of the derivative, and a sum or a difference is differentiated term by term:

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

\displaystyle =

\displaystyle c\cdot f'(x)

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

\displaystyle =

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

The example below differentiates $\displaystyle {f(x)}={4x^3-5x^2+7x-2}$ term by term:

f'(x)=4\cdot3x^2-5\cdot2x+7\cdot1-0=12x^2-10x+7

A fraction in front stays in front too: $\displaystyle {\left(\frac{1}{3}x^3\right)'}={\frac{1}{3}\cdot3x^2}={x^2}$ .

A product: expanding before differentiating

The derivative of a product is not the product of the derivatives. For now, expand the brackets, and only then differentiate.
$\displaystyle {f(x)}={(x+3)(2x-1)}$ .

f(x)=2x^2+5x-3

f'(x)=4x+5

The product of the derivatives, $\displaystyle {(x+3)'\cdot(2x-1)'}={1\cdot2}={2}$ , is not the derivative.
A rule for a product, without expanding, comes in the next lesson.

The derivative at a point

To know the slope of the tangent at a point, put its $\displaystyle x$ into the derivative function.
$\displaystyle {f(x)}={x^3-3x^2+2}$ .
The example below finds $\displaystyle f'(x)$ , and puts $\displaystyle {x}={1}$ and $\displaystyle {x}={2}$ into it.

f'(x)=3x^2-6x

f'(1)=3-6=-3,\qquad f'(2)=12-12=0

On the board: $\displaystyle f$ , and the tangents at the points $\displaystyle (1,0)$ and $\displaystyle (2,-2)$ (the dashed ones).

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

\displaystyle y=-2

\displaystyle y=-3x+3

At $\displaystyle {x}={1}$ the slope of the tangent is $\displaystyle -3$ : the graph falls there.
At $\displaystyle {x}={2}$ the slope of the tangent is 0: the tangent is level, $\displaystyle {y}={-2}$ .
To find where the slope is a given number, solve an equation: $\displaystyle {f'(x)}={0}$ , that is $\displaystyle {3x^2-6x}={0}$ , gives $\displaystyle {x}={0}$ or $\displaystyle {x}={2}$ .

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

Expand the brackets, and then differentiate.

/ – 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

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

Find $\displaystyle f'(x)$ .

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

/ – 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

Work out $\displaystyle f'(2)$ and $\displaystyle f'(0)$ .

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

/ – 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

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

/ – 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

At which of the points $\displaystyle (0,0)$ and $\displaystyle (2,0)$ is the tangent to the graph of $\displaystyle f$ level?
What is the slope of the tangent at the other point?

Exercise 4

$\displaystyle {f(x)}={x^2-6x}$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=x^{2}-6x$

/ – 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

At which point is the slope of the tangent to the graph of $\displaystyle f$ equal to 0?
Write an equation and solve it.

At which point is the slope of the tangent to the graph of $\displaystyle f$ equal to 4?
Write an equation and solve it.

Exercise 5

$\displaystyle {g(x)}={x^3-12x}$ .

Find $\displaystyle g'(x)$ .

$\displaystyle g(x)=x^{3}-12x$

/ – 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

At which points is the tangent to the graph of $\displaystyle g$ level?
Write an equation and solve it.

Exercise 6

A stone falls from a roof.
$\displaystyle {s(t)}={5t^2}$ – the distance in metres the stone falls in the first $\displaystyle t$ seconds.
The stone’s speed at the moment $\displaystyle t$ is $\displaystyle s'(t)$ .
What is the stone’s speed after 3 seconds?

$\displaystyle s(t)=5t^{2}$

/ – 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: 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
$\displaystyle f'(x)=0$