Differentiation in Core 3 (C3) are an extension of the work that you did in Core 1 and Core 2.

Differentiation

Standard Derivatives

For the C3 module, there are a few standard results for differentiation that need to be learnt. These are:

$\frac{d}{d x} \ln x = \frac{1}{x}$

$\frac{d}{d x} e^{k x} = k e^{k x}$

$\frac{d}{d x} \sin k x = k \cos k x$

$\frac{d}{d x} \cos k x = - k \sin k x$

$\frac{d}{d x} \tan k x = \frac{k}{c o s^{2} k x}$

Chain Rule

$\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x}$

The Chain Rule is used to differentiate when one function is applied to another function. A typical example of this is:

$y = \sin (x^{2})$

One of the ways of remembering the chain rule is: Find the derivative outside, then multiply it by the derivative inside. In the example above, this becomes:

$\frac{d y}{d x} = 2 x \cos (x^{2})$

Product Rule

$\frac{d}{d x} u v = v \frac{d u}{d x} + u \frac{d v}{d x}$

The product rule is used when two functions are multiplied together.

Quotient Rule

$\frac{d}{d x} \frac{u}{v} = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}$

The quotient rule is used when one function is divided by another. It is a specific case of the product rule. A typical example of this is:

Implicit Differentiation

Implicit differentiation is used when a function is not a simple $y = s o m e t h i n g$ but contains a mixture of x and y parts. A typical example of this is to differentiate:

$y^{2} + 2 y = 4 x^{3}$

When differentiating the y components of the expression you differentiate as normal, and then multiply by $\frac{d y}{d x}$ . So differentiating both sides of the above expression it becomes:

$2 y \frac{d y}{d x} + 2 \frac{d y}{d x} = 12 x^{2}$

Then by factorising the left hand side and cancelling, this becomes:

$\frac{d y}{d x} = \frac{6 x^{2}}{y + 1}$