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)
\begin{equation} f'(x)=g'(x)+h'(x) \end{equation}

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)
\begin{equation} f'(x)=g(x)h'(x) + h(x) g'(x) \end{equation}

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)
\begin{equation} f'(x)=g'(h(x)) h'(x) \end{equation}

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)
\begin{equation} y=g(u) \end{equation}

with

(9)
\begin{equation} u=f(x) \end{equation}

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.

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page revision: 10, last edited: 28 Sep 2011 20:19