A-level Mathematics/MEI/C3/Differentiation

From Wikibooks, open books for an open world
< A-level Mathematics‎ | MEI‎ | C3
Jump to: navigation, search

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


Standard Derivatives[edit]

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[edit]

\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[edit]

\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[edit]

\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[edit]

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}