New Zealand
Level 7 - NCEA Level 2

# Derivative of a polynomial (expansion then power rule)

Lesson

Sometimes polynomials are in factored form.

For example $f(x)=(x+3)(x-1)$f(x)=(x+3)(x1), or $g(x)=x(x+1)^2$g(x)=x(x+1)2

In there present form we are unable to use the power rule to find the derivative.  But, we can with relative ease, fully expand the functions which will leave us with individual terms that we can use.

##### Example 1

$f(x)=(x+3)(x-1)$f(x)=(x+3)(x1)

$f(x)=x^2+3x-x-3=x^2+2x-3$f(x)=x2+3xx3=x2+2x3

so $f'(x)=2x+2$f(x)=2x+2

##### Example 2

$g(x)=x(x+1)^2$g(x)=x(x+1)2

$g(x)=x(x^2+2x+1)=x^3+2x^2+x$g(x)=x(x2+2x+1)=x3+2x2+x

So $g'(x)=3x^2+4x+1$g(x)=3x2+4x+1

#### Worked Examples

##### QUESTION 1

Consider the function $y=\left(x+4\right)^2$y=(x+4)2

1. Express the function $y$y in expanded form.

2. Hence find the derivative $\frac{dy}{dx}$dydx of the function $y=\left(x+4\right)^2$y=(x+4)2

##### QUESTION 2

Consider the function $f\left(x\right)=\left(\sqrt{x}+10x^2\right)^2$f(x)=(x+10x2)2

1. Express the function $f\left(x\right)$f(x) in expanded form, with all terms written as powers of $x$x.

2. Hence find the derivative $f'\left(x\right)$f(x) of the function $f\left(x\right)=\left(\sqrt{x}+10x^2\right)^2$f(x)=(x+10x2)2

Remove all fractional indices in your final answer.

##### QUESTION 3

By first expanding, differentiate the function $f\left(x\right)=\left(x+2\right)^3$f(x)=(x+2)3

### Outcomes

#### M7-10

Apply differentiation and anti-differentiation techniques to polynomials

#### 91262

Apply calculus methods in solving problems