NZ Level 8 (NZC) Level 3 (NCEA) [In development]
Mixed Problems
Lesson

This set is for you to practice and identify which rule is most effective based on the presentation of the functions given.

We have looked at 4 rules so far, let's recap.

## Power Rule

Used to differentiate individual terms.

Power Rule for $ax^n$axn!

For a function $f(x)=ax^n$f(x)=axn, the derivative $f'(x)=nax^{n-1}$f(x)=naxn1

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

When $a=1$a=1, we get the same rule as before.  So really this is the only one we need to remember.

## Chain Rule

Used to differentiate powers of a function.

Chain Rule

If $y=f(u)$y=f(u) and $u=g(x)$u=g(x) then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$dydx=dydu·dudx

In words you can remember this little rhythm.

“ the derivative of the outside, times the derivative of the inside”

## Product Rule

Used to differentiate functions that are comprised of parts being multiplied together.

Product Rule

If $y=u\times v$y=u×v then $y'=uv'+vu'$y=uv+vu

If $y=u(x)v(x)$y=u(x)v(x) then $\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$dydx=udvdx+vdudx

In words you can remember this little rhythm.

The first times the derivative of the second plus the second times the derivative of the first.

## Quotient Rule

Used to differentiate functions that are a quotient (a division of one piece by another).

Quotient Rule

If $y=\frac{u}{v}$y=uv and u and v are both functions of the same variable, then

$y'=\frac{vu'-uv'}{v^2}$y=vuuvv2

Be very careful with the subtraction in the numerator - this often creates and expansion with a negative coefficient that often leads to student errors.

Take them slowly, step by step, setting out as much work as possible, so that errors (if you make them) are easier to identify.

#### Worked Examples

##### QUESTION 1

Differentiate $y=x^3\left(5x+3\right)^7$y=x3(5x+3)7 using the product rule. Express your answer in factorised form.

You may let $u=x^3$u=x3 and $v=\left(5x+3\right)^7$v=(5x+3)7.

##### QUESTION 2

Find the derivative of $y=\left(5x\right)^3+6\sqrt[3]{x}$y=(5x)3+63x.

##### QUESTION 3

Consider the function $f\left(t\right)=\frac{\left(4t^2+3\right)^3}{\left(5+2t\right)^5}$f(t)=(4t2+3)3(5+2t)5.

Find $f'\left(t\right)$f(t).

You may use the substitutions $u=\left(4t^2+3\right)^3$u=(4t2+3)3 and $v=\left(5+2t\right)^5$v=(5+2t)5 in your working.

### Outcomes

#### M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

#### 91578

Apply differentiation methods in solving problems