The derivative of a root function

Calculus, grade 11 (4 units) Lesson 11 of 20Learning

The derivative of a root function

A reminder from grade 10

The derivative of the square root of a function $\displaystyle u(x)$ is the derivative of $\displaystyle u$ divided by twice the root:

\displaystyle \left(\sqrt{u}\right)'

\displaystyle =

\displaystyle \frac{u'}{2\sqrt{u}}

$\displaystyle {\left(\sqrt{x^2+5}\right)'}={\frac{2x}{2\sqrt{x^2+5}}}={\frac{x}{\sqrt{x^2+5}}}$ .
A function times a root is differentiated by the product rule: $\displaystyle {(u\cdot v)'}={u'\cdot v+u\cdot v'}$ .

In this lesson $\displaystyle f$ and $\displaystyle g$ are polynomials of degree at most 2, and the functions are $\displaystyle {y}={\sqrt{\frac{f(x)}{g(x)}}}$ and $\displaystyle {y}={g(x)\cdot\sqrt{f(x)}}$ .

The domain

A square root is defined only for a number that is not negative, and a fraction only when its denominator is not 0:

$\displaystyle {y}={g(x)\cdot\sqrt{f(x)}}$ is defined where $\displaystyle f(x)\geq0$ .
$\displaystyle {y}={\sqrt{\frac{f(x)}{g(x)}}}$ is defined where $\displaystyle \frac{f(x)}{g(x)}\geq0$ – an inequality of a fraction, solved as in the lesson on positive and negative values.

For example, $\displaystyle {y}={\sqrt{\frac{x-1}{x+2}}}$ : the fraction is positive or 0 when $\displaystyle x<-2$ or $\displaystyle x\geq1$ . At $\displaystyle {x}={1}$ the numerator is 0, and the root is $\displaystyle 0$ ; at $\displaystyle {x}={-2}$ the denominator is 0, so $\displaystyle -2$ is not in the domain.

The root of a fraction

$\displaystyle {y}={\sqrt{\frac{x-1}{x+2}}}$ .
In the example below we differentiate by the root rule with $\displaystyle {u}={\frac{x-1}{x+2}}$ , and find $\displaystyle u'$ by the quotient rule:

$\displaystyle {u'}={\frac{1\cdot(x+2)-(x-1)\cdot1}{(x+2)^2}}={\frac{3}{(x+2)^2}}$ .
$\displaystyle {y'}={\frac{u'}{2\sqrt{u}}}={\frac{3}{2(x+2)^2\sqrt{\frac{x-1}{x+2}}}}$ .

The derivative is defined on the domain except at $\displaystyle {x}={1}$ : there the root in the denominator is 0.

A polynomial times a root

$\displaystyle {y}={x\sqrt{4-x}}$ , and its domain is $\displaystyle x\leq4$ .
In the example below we differentiate by the product rule, and write the derivative as one fraction:

y'=1\cdot\sqrt{4-x}+x\cdot\frac{-1}{2\sqrt{4-x}}=\frac{2(4-x)-x}{2\sqrt{4-x}}=\frac{8-3x}{2\sqrt{4-x}}

To add them, $\displaystyle \sqrt{4-x}$ is multiplied above and below by $\displaystyle 2\sqrt{4-x}$ : $\displaystyle {\sqrt{4-x}\cdot\sqrt{4-x}}={4-x}$ .

A tangent

In the example below we find the equation of the tangent to the graph of $\displaystyle {y}={x\sqrt{4-x}}$ at the point where $\displaystyle {x}={3}$ :

The point: $\displaystyle {y(3)}={3\sqrt{1}}={3}$ , that is $\displaystyle (3,3)$ .
The slope: $\displaystyle {m}={y'(3)}={\frac{8-9}{2\sqrt{1}}}={-\frac{1}{2}}$ .
The tangent: $\displaystyle {y-3}={-\frac{1}{2}(x-3)}$ , that is $\displaystyle {y}={-\frac{1}{2}x+\frac{9}{2}}$ .

\displaystyle f(x)=x\sqrt{4-x}

\displaystyle y=-\frac{1}{2}x+\frac{9}{2}

\displaystyle (3,3)

Exercise 1

$\displaystyle {y}={\sqrt{\frac{x+3}{x-2}}}$ .
Find the domain: solve the inequality.

Exercise 2

Differentiate, and write the derivative as one fraction.

/ – 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 3

Work out $\displaystyle f'(3)$ .

$\displaystyle f(x)=\sqrt{\frac{x^{2}+3}{x+1}}$

/ – 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

$\displaystyle {f(x)}={(x-2)\sqrt{x}}$ .
The point $\displaystyle (4,4)$ is on the graph of $\displaystyle f$ .

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

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

/ – 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

Write the equation of the tangent on the board.

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

\displaystyle (4,4)

Formulas: calculus, grade 11 (4 units)

Open as a page

Transformations of the graph of $f$

A horizontal shift: $\displaystyle p$ units to the right ( $\displaystyle p>0$ ) or to the left ( $\displaystyle p<0$ )
$\displaystyle y=f(x-p)$
A vertical shift: $\displaystyle k$ units up ( $\displaystyle k>0$ ) or down ( $\displaystyle k<0$ )
$\displaystyle y=f(x)+k$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$
A vertical stretch ( $\displaystyle a>1$ ) or squeeze ( $\displaystyle 0<a<1$ )
$\displaystyle y=a\cdot f(x)$
Absolute value: the part below the $\displaystyle x$ -axis is reflected above it
$\displaystyle y=|f(x)|$

The rational function

The horizontal asymptote when the numerator and the denominator have the same degree – $\displaystyle a$ and $\displaystyle b$ the coefficients of the highest power
$\displaystyle y=\frac{a}{b}$
The horizontal asymptote when the numerator’s degree is lower than the denominator’s
$\displaystyle y=0$
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
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$

Derivatives and the tangent

The derivative of a power
$\displaystyle \left(x^{n}\right)'=nx^{n-1}$
The derivative of a product
$\displaystyle \left(u\cdot v\right)'$ $\displaystyle =$ $\displaystyle u'\cdot v+u\cdot v'$
The derivative of a quotient
$\displaystyle \left(\frac{u}{v}\right)'$ $\displaystyle =$ $\displaystyle \frac{u'\cdot v-u\cdot v'}{v^{2}}$
The derivative of $\displaystyle \frac{c}{x}$
$\displaystyle \left(\frac{c}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{c}{x^{2}}$
The derivative of a root – $\displaystyle u$ a function of $\displaystyle x$
$\displaystyle \left(\sqrt{u}\right)'$ $\displaystyle =$ $\displaystyle \frac{u'}{2\sqrt{u}}$
The equation of the tangent at the point $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1$ $\displaystyle =$ $\displaystyle m(x-x_1)$
The slope of the tangent
$\displaystyle m$ $\displaystyle =$ $\displaystyle f'(x_1)$

Investigating a function with its derivative

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A possible extremum point – check the sign of $\displaystyle f'$ on each side of it
$\displaystyle f'(x)=0$

Integrals and areas

The integral of a power
$\displaystyle \int x^{n}\,dx$ $\displaystyle =$ $\displaystyle \frac{x^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$
The integral of $\displaystyle \frac{c}{(ax+b)^{2}}$
$\displaystyle \int\frac{c}{(ax+b)^{2}}\,dx$ $\displaystyle =$ $\displaystyle -\frac{c}{a(ax+b)}+C$
The definite integral – $\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx$ $\displaystyle =$ $\displaystyle F(b)-F(a)$
The area between two graphs, from $\displaystyle {x}={a}$ to $\displaystyle {x}={b}$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$