topic badge
India
Class XI

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

11.C.LD.1

Derivative introduced as rate of change both as that of distance function and geometrically, intuitive idea of limit. Definition of derivative, relate it to slope of tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.

What is Mathspace

About Mathspace