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:

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