NZ Level 7 (NZC) Level 2 (NCEA)
Derivative of a Sum (ax^n)
Lesson
Just like we saw in derivatives of a sum of the type x^n,

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.

And this applies to any function, whether it be the $x^n$xn we saw before, or $ax^n$axn that we are exploring now.

#### Examples

Find the derivative of the following,

a) $f(x)=4x^2+3x+2$f(x)=4x2+3x+2, then $f'(x)=8x+3$f(x)=8x+3   (remember that the derivative of a constant term is $0$0)

b) $f(x)=3x^3-3x^2$f(x)=3x33x2,  then $f'(x)=9x^2-6x$f(x)=9x26x

c) $f(x)=6x^{-3}-2x+\sqrt{x}$f(x)=6x32x+x.  Firstly we need to turn the $\sqrt{x}$x into a power.  $\sqrt{x}=x^{\frac{1}{2}}$x=x12 So $f(x)=6x^{-3}-2x+x^{\frac{1}{2}}$f(x)=6x32x+x12 and so then the derivative $f'(x)=-18x^{-4}-2+\frac{1}{2}x^{-\frac{1}{2}}$f(x)=18x42+12x12

#### Worked Examples:

##### question 1

Differentiate $y=7x^2-9x+8$y=7x29x+8.

##### question 2

Differentiate $y=\frac{24}{x^5}-\frac{30}{x^4}$y=24x530x4.

##### question 3

Find the derivative of $y=x^3\sqrt{x}+3x^5$y=x3x+3x5.

### Outcomes

#### M7-10

Apply differentiation and anti-differentiation techniques to polynomials

#### 91262

Apply calculus methods in solving problems