Derivatives Of More Complex Functions

### Slightly More Complex Functions

The SL syllabus does not require students to be able to derive the following derivatives. They are on the formula handbook.

 Function Derivative $x^n$ $nx^{n-1}$ $sin(x)$ $cos(x)$ $cos(x)$ $-sin(x)$ $tan(x)$ $\frac{1}{cos^2(x)}$ $e^x$ $e^x$ $ln(x)$ $\frac{1}{x}$

The HL folk will need to know how to derive some or all of these… These derivatives can be proved using some clever tricks. Check out Google for more details…

### Derivative "Rules"

Once you know the derivatives of a few basic functions the math teachers turn up the heat and start asking more complex questions such as find the derivative of $f(x) = 3x^2+x + 3$ or $g(x)=x sin^2(x)$… the possibilities are almost endless. There are a few basic rules that allow you to deal with these more complex functions without having to resort to using "first principles."

#### Sum/Difference Rule

A function can be defined as the sum (or difference) of two or more other functions, such as $f(x)=g(x)+h(x)$. It does not take too much work to show from first principles that the derivative of this function is:

(1)
$$f'(x)=g'(x)+h'(x)$$

or in Leibniz notation:

(2)
\begin{align} \frac{d}{dx} f(x) = \frac{d}{dx}g(x) + \frac{d}{dx} h(x) \end{align}

This is not on the IB formula booklet, but we all hope you can remember.

#### Product Rule

A function can be defined as the product of two (or more) functions, such as $f(x)=g(x)h(x)$. With some work and cleverness the derivative of the product of two functions can be shown to be:

(3)
$$f'(x)=g(x)h'(x) + h(x) g'(x)$$

or in Leibniz notation:

(4)
\begin{align} \frac{d}{dx}f(x)=g(x)\frac{d}{dx} h(x) + h(x) \frac{d}{dx} g(x) \end{align}

#### Quotient Rule

A function can be defined as the quotient of two functions, such as $f(x)=\frac{g(x)}{h(x)}]$. The derivative of this type of function can be shown to be:

(5)
\begin{align} f'(x)=\frac{h(x) g'(x) - g(x) h'(x)}{(h(x))^2} \end{align}

or in Leibniz notation:

(6)
\begin{align} \frac{d}{dx}f(x)= \frac{h(x) \frac{d}{dx}g(x) - g(x) \frac{d}{dx}h(x)}{(h(x))^2} \end{align}

#### Chain Rule

The last one is often found to be the toughest. This involves finding the derivative of a composite function. For example the function $f(x)=g(h(x))$. The derivative of a composite function is:

(7)
$$f'(x)=g'(h(x)) h'(x)$$

In words that means the derivative is equal to the derivative of the "outside" function evaluated at the "inside" function times the derivative of the inside function.

Yeah that's a mouth full. In Leibniz notation the Chain rule can look very different. If we define:

(8)
$$y=g(u)$$

with

(9)
$$u=f(x)$$

Notice then that $y=g(f(x))$ just the good old composite function. Then we can say that:

(10)
\begin{align} \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx} \end{align}

I frankly find them both a bit confusing. It just takes a bit of practice to deal with either way of writing the rule.

#### Examples

Any suggestions for example problems? Post them in the comments below.