Differentiation

New Zealand

Level 7 - NCEA Level 2

Lesson

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.

$f(x)=(x+3)(x-1)$`f`(`x`)=(`x`+3)(`x`−1)

$f(x)=x^2+3x-x-3=x^2+2x-3$`f`(`x`)=`x`2+3`x`−`x`−3=`x`2+2`x`−3

so $f'(x)=2x+2$`f`′(`x`)=2`x`+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`(`x`2+2`x`+1)=`x`3+2`x`2+`x`

So $g'(x)=3x^2+4x+1$`g`′(`x`)=3`x`2+4`x`+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 $\frac{dy}{dx}$

`d``y``d``x` of the function $y=\left(x+4\right)^2$`y`=(`x`+4)2

Consider the function $f\left(x\right)=\left(\sqrt{x}+10x^2\right)^2$`f`(`x`)=(√`x`+10`x`2)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`+10`x`2)2Remove all fractional indices in your final answer.

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