S-Cool Revision Summary

S-Cool Revision Summary

 

Differentiation

Differentiation is used to find the gradient at a point on a curve

The basic rules are:

y = f(x)

xn

nxn-1

ex

ex

ln x

sin x

cos x

cos x

-sin x

tan x

sec2 x

When an expression has several terms, linked by addition and subtraction, differentiate each term separately.

Uses of Differentiation

To find equations of tangents and normals to the curve:

  1. Differentiate the function.

  2. Put in the x-value into to find the gradient of the tangent.

  3. Put in the x-value into the function (y = ...) to find the coordinates of the point where the tangent touches the curve.

Put these values into the formula for a straight line:

y - y1 = m (x - x1), where m = gradient and (x1, y1) is where the tangent meets the curve.

The gradient of the normal is -1 times the gradient of the tangent (as they are perpendicular).

Stationary Points occur when the gradient = 0.

A stationary point will be one of three types: a Maximum, a Minimum or an Inflection.

Determine the nature of a stationary point using one of these methods.

  1. Already knowing the shape of the graph.

  2. By looking at the gradient either side of the stationary point.

  3. By using the second derivative, .

For a particular value for x, when = 0,

If > 0 there is a minimum.

If < 0 there is a maximum.

If = 0 there could be a maximum, minimum or an inflection.

The Chain Rule

Copyright S-cool

  1. Use a substitution to turn the function into one you can differentiate easily.

  2. Differentiate the two functions you now have.

  3. Multiply them together, leaving your answer in terms of the original question (i.e. in y's and x's).

When accomplished this can be done in one go, by inspection, resulting in these rules:

Function

Derivative

y =

=

y =

=

y =

=

y =

=

y =

=

y =

=

The Product Rule and The Quotient Rule

The product rule states that for two functions, u and v,

If y = uv, then = .

After using the product rule you will normally be able to factorise the derivative.

The quotient rule states that for two functions, u and v,

If then = .