Derivative of a polynomial (expansion then power rule)
Sometimes polynomials are in factored form.
For example $f(x)=(x+3)(x-1)$f(x)=(x+3)(x−1), 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)(x−1)
$f(x)=x^2+3x-x-3=x^2+2x-3$f(x)=x2+3x−x−3=x2+2x−3
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
Express the function $y$y in expanded form.
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
Express the function $f\left(x\right)$f(x) in expanded form, with all terms written as powers of $x$x.
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