New Zealand
Level 7 - NCEA Level 2

# Derivative of a Sum (x^n)

Lesson

We just looked a our first of many rules for differentiation (the process of finding the derivative).

Power Rule for $x^n$xn!

For a function $f(x)=x^n$f(x)=xn, the derivative $f'(x)=nx^{n-1}$f(x)=nxn1

$n$n can be positive or negative, integer or fraction

Adding together terms in varying power forms create other functions, which in turn we can find the derivative of.

For example if we have the functions $h(x)=2x$h(x)=2x and $g(x)=x^3$g(x)=x3 then we can add $h(x)$h(x) and $g(x)$g(x) together to create a new function $f(x)=x^2+2x$f(x)=x2+2x

To see what happens with the sum of the function and the resulting gradient function, let's look at 3 parts. Firstly $h(x)$h(x), then $g(x)$g(x) and then $f(x)$f(x).

$h(x)=2x$h(x)=2x then $h'(x)=2$h(x)=2

for $g(x)=x^3$g(x)=x3 then $g'(x)=3x^2$g(x)=3x2

and finally for $f(x)=x^3+2x$f(x)=x3+2x

Well, we are not sure yet how to find this so for the last one we will use first principles.

So we need to know $f(x)$f(x) and $f(x+h)$f(x+h)

$f(x)=x^3+2x$f(x)=x3+2x

$f(x+h)=(x+h)^3+2(x+h)$f(x+h)=(x+h)3+2(x+h)

$f(x+h)=h^3+3h^2x+3hx^2+x^3+2x+2h$f(x+h)=h3+3h2x+3hx2+x3+2x+2h

$f(x+h)=x^3+2x+h^3+3h^2x+3hx^2+2h$f(x+h)=x3+2x+h3+3h2x+3hx2+2h

$f(x+h)=[h^3+3h^2x+3hx^2+2h]+[x^3+2x]$f(x+h)=[h3+3h2x+3hx2+2h]+[x3+2x]

 $\frac{dy}{dx}$dydx​ $=$= $\frac{dy}{dx}$dydx​ $=$= $\frac{dy}{dx}$dydx​ $=$= $\frac{dy}{dx}$dydx​ $=$= $\frac{dy}{dx}$dydx​ $=$= $\frac{dy}{dx}$dydx​ $=$= $3x^2+2$3x2+2

So we can see that the derivative of the sum is the same as the sums of the derivatives.

Sum of Derivatives

Derivative of sum is equal to the sum of the derivatives.

If $f(x)=g(x)\pm h(x)$f(x)=g(x)±h(x) then $f'(x)=g'(x)\pm h'(x)$f(x)=g(x)±h(x)

This means we can apply the power rule to individual terms.

#### Examples

Find the derivative of the following,

a) $f(x)=x^2+x+2$f(x)=x2+x+2, then $f'(x)=2x+1$f(x)=2x+1   (remember that the derivative of a constant term is $0$0)

b) $f(x)=x^3-x^2$f(x)=x3x2,  then $f'(x)=3x^2-2x$f(x)=3x22x

c) $f(x)=x^{-3}-x+\sqrt{x}$f(x)=x3x+x.  Firstly we need to turn the $\sqrt{x}$x into a power.  $\sqrt{x}=x^{\frac{1}{2}}$x=x12 So $f(x)=x^{-3}-x+x^{\frac{1}{2}}$f(x)=x3x+x12 and so then the derivative $f'(x)=-3x^{-4}-1+\frac{1}{2}x^{-\frac{1}{2}}$f(x)=3x41+12x12

#### Worked Examples:

##### question 1

Determine the derivative of $y=x^4+x^5$y=x4+x5.

##### question 2

Find the derivative of $y=x^3+x^{-5}-9$y=x3+x59.

##### question 3

Find the derivative of $y=\frac{x+1}{\sqrt[7]{x}}$y=x+17x.

### Outcomes

#### M7-10

Apply differentiation and anti-differentiation techniques to polynomials

#### 91262

Apply calculus methods in solving problems