A-level Mathematics/MEI/C3/Differentiation

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:

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

$\frac{d}{dx}{e}^{kx}=k{e}^{kx}$

$\frac{d}{dx}\sin kx=k\cos kx$

$\frac{d}{dx}\cos kx=-k\sin kx$

$\frac{d}{dx}\tan kx=\frac{k}{co{s}^{2}kx}$

Chain Rule

$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$

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:

$\frac{dy}{dx}=2x\cos (x^2)$

Product Rule

$\frac{d}{dx}uv=v\frac{du}{dx}+u\frac{dv}{dx}$

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

Quotient Rule

$\frac{d}{dx}\frac{u}{v}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$

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 $y=something$ but contains a mixture of x and y parts. A typical example of this is to differentiate:

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

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

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

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

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