The derivative of the square root function

Calculus, grade 10 (5 units) Lesson 19 of 26Learning

The derivative of the square root function

The derivative of

\displaystyle \sqrt{x}

from the definition

The derivative of $\displaystyle \sqrt{x}$ is $\displaystyle \frac{1}{2\sqrt{x}}$ . The proof below writes the average rate of change of $\displaystyle \sqrt{x}$ from $\displaystyle x$ to $\displaystyle x+h$ , multiplies the numerator and the denominator by the conjugate $\displaystyle \sqrt{x+h}+\sqrt{x}$ , and shows which number the quotient approaches as $\displaystyle h\to0$ .

\frac{\sqrt{x+h}-\sqrt{x}}{h}=\frac{(\sqrt{x+h}-\sqrt{x})(\sqrt{x+h}+\sqrt{x})}{h(\sqrt{x+h}+\sqrt{x})}

The numerator is a difference of two squares: $\displaystyle {(\sqrt{x+h})^2-(\sqrt{x})^2}={(x+h)-x}={h}$ .

=\frac{h}{h(\sqrt{x+h}+\sqrt{x})}=\frac{1}{\sqrt{x+h}+\sqrt{x}}

As $\displaystyle h\to0$ , $\displaystyle \sqrt{x+h}\to\sqrt{x}$ , so the denominator approaches $\displaystyle 2\sqrt{x}$ .

\displaystyle (\sqrt{x})'

\displaystyle =

\displaystyle \frac{1}{2\sqrt{x}}

\displaystyle x>0

The derivative by the reflection

$\displaystyle \sqrt{x}$ is the reflection of $\displaystyle x^2$ (where $\displaystyle x\geq0$ ) in the line $\displaystyle {y}={x}$ , so their tangents are reflections of each other too. A reflection swaps the “step right” and the “step up”, so the slope of the reflected tangent is the reciprocal of the original slope.

On the graph below:

The tangent to $\displaystyle {y}={x^2}$ at the point $\displaystyle (2,4)$ has the slope $\displaystyle {2\cdot2}={4}$ .
The tangent to $\displaystyle {y}={\sqrt{x}}$ at the point $\displaystyle (4,2)$ is its reflection, and its slope is $\displaystyle \frac{1}{4}$ .

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

\displaystyle g(x)=\sqrt{x}

\displaystyle y=x

\displaystyle y=4x-4

\displaystyle y=\frac{1}{4}x+1

\displaystyle A=(2,4)

\displaystyle B=(4,2)

In general, at the point $\displaystyle (a,a^2)$ the slope of the tangent to the parabola is $\displaystyle 2a$ , and at the reflected point $\displaystyle (a^2,a)$ the slope of the tangent to the root is $\displaystyle \frac{1}{2a}$ . Here $\displaystyle {x}={a^2}$ , that is $\displaystyle {a}={\sqrt{x}}$ – and again we get $\displaystyle \frac{1}{2\sqrt{x}}$ .

A vertical tangent at the origin

At $\displaystyle {x}={0}$ , $\displaystyle \sqrt{x}$ has no derivative: $\displaystyle \frac{1}{2\sqrt{0}}$ is a division by 0. The graph shows why:

The tangent to the parabola at the origin is the $\displaystyle x$ -axis, with slope 0.
Its reflection is the $\displaystyle y$ -axis: a vertical tangent, and a vertical line has no slope.
As $\displaystyle x$ gets close to 0, $\displaystyle \frac{1}{2\sqrt{x}}$ grows without limit – the graph gets steeper and steeper.

The root of a function

By the chain rule, the derivative of $\displaystyle \sqrt{f(x)}$ is the outer derivative, $\displaystyle \frac{1}{2\sqrt{f(x)}}$ , times the inner derivative, $\displaystyle f'(x)$ :

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

\displaystyle =

\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}

\displaystyle f(x)>0

$\displaystyle {(\sqrt{ax+b})'}={\frac{a}{2\sqrt{ax+b}}}$ . For example $\displaystyle {(\sqrt{4x-3})'}={\frac{4}{2\sqrt{4x-3}}}={\frac{2}{\sqrt{4x-3}}}$ .
$\displaystyle {(\sqrt{x^2+16})'}={\frac{2x}{2\sqrt{x^2+16}}}={\frac{x}{\sqrt{x^2+16}}}$ .

The tangent to the graph of a root function

The tangent is found as for a polynomial: a point and a slope. The example below finds the tangent to the graph of $\displaystyle {f(x)}={\sqrt{x}}$ at the point where $\displaystyle {x}={4}$ :

$\displaystyle {f(4)}={\sqrt{4}}={2}$ : the point of contact $\displaystyle (4,2)$ .
$\displaystyle {f'(4)}={\frac{1}{2\sqrt{4}}}={\frac{1}{4}}$ : the slope of the tangent.
$\displaystyle {y-2}={\frac{1}{4}(x-4)}$ , that is $\displaystyle {y}={\frac{1}{4}x+1}$ .

\displaystyle f(x)=\sqrt{x}

\displaystyle y=\frac{1}{4}x+1

\displaystyle A=(4,2)

Exercise 1

$\displaystyle {f(x)}={\sqrt{x}}$ .
The average rate of change of $\displaystyle f$ from $\displaystyle {x}={9}$ to $\displaystyle {x}={9+h}$ is $\displaystyle \frac{\sqrt{9+h}-3}{h}$ .

Multiply the numerator and the denominator by the conjugate $\displaystyle \sqrt{9+h}+3$ , and simplify.

/ – 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

Which number does the quotient approach as $\displaystyle h\to0$ ?
What is $\displaystyle f'(9)$ ? Check by the formula $\displaystyle \frac{1}{2\sqrt{x}}$ .

/ – 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

Exercise 2

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

The function is $\displaystyle {f(x)}={\sqrt{2x+1}}$ .
Work out $\displaystyle f'(4)$ .

$\displaystyle f(x)=\sqrt{2x+1}$

/ – 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 is $\displaystyle {f(x)}={\sqrt{x+5}}$ .
The point $\displaystyle (4,3)$ is on its graph.

Work out the slope of the tangent to the graph at the point $\displaystyle (4,3)$ .

$\displaystyle f(x)=\sqrt{x+5}$

/ – 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 in a row of your own on the board.

\displaystyle f(x)=\sqrt{x+5}

\displaystyle (4,3)

Exercise 5

The function is $\displaystyle {f(x)}={\sqrt{2x-1}}$ .
At the point $\displaystyle A$ on the graph the slope of the tangent is $\displaystyle \frac{1}{3}$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\sqrt{2x-1}$

/ – 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: what is the $\displaystyle x$ -coordinate of $\displaystyle A$ ?

Write on the board the equation of the tangent at the point $\displaystyle A$ .

\displaystyle f(x)=\sqrt{2x-1}

Exercise 6

The function is $\displaystyle {f(x)}={\sqrt{x-2}}$ .
At which point on the graph is the tangent vertical?
Explain why $\displaystyle f$ has no derivative at that point.

Formulas: calculus, grade 10 (5 units)

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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
$\displaystyle y=b$