Differentiation Revision for IB SL

Rule used to differentiate Type of function Rule used to differentiate Polynomial Constant Always becomes zero Remember that , e , ln(3), are still constants Composite function (function of a function) Chain rule

Rule used to differentiate Type of function Rule used to differentiate 2 functions of x multiplied together Product rule 1 function of x divided by another function of x Quotient rule Exponential function

Rule used to differentiate Type of function Rule used to differentiate Natural logarithm Trigonometric functions

Chain Rule

Product Rule

Quotient Rule

ex

ln(x)

Trigonometric functions

Find gradient at particular point Substitute the x-value for that point into the expression for dy/dx. With implicit functions, you will need the x and y values.

Find tangent or normal A tangent or normal is a straight line (y = mx + c). For the tangent, m is the gradient at that point. For the normal, m is

Find turning points and their nature (min/max) Turning pt is when dy/dx=0 To determine nature: Use 2nd derivative test If d2y/dx2: > 0, local minimum < 0, local maximum = 0, test inconclusive – could be min, max or point of inflection Examine gradient on either side of the point. Use this method if finding the 2nd derivative is too hard or if the test was inconclusive.

Find turning points and their nature (min/max)

Points of inflection A point of inflection is the point on a curve when it changes from concave-up to concave-down, or vice-versa. This is the point when the second derivative, d2y/dx2, equals zero

Summary

Motion of particles s, displacement v, velocity a, acceleration E.g. what is the velocity function for a particle if its displacement function is s = (3x-2)4 Differentiate