### 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)or in Leibniz notation:

(2)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)or in Leibniz notation:

(4)#### 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)or in Leibniz notation:

(6)#### 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)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)with

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

(10)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.

Hello, just a quick question.

For chain rule you said that the composite function was f(x)=g(h(x)),

but later on you said that the function was y=g(f(x)). What is the actual function?

Thank you for the notes they are really helpful.

ReplyOptionsf(x)=g(h(x)), is an example of a composite function.

y=g(f(x)) is another composite function which was used for explanation later on.

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