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:

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


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


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


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


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

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

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

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:

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