implicit differentiation

derivative plotter

this applet enables you to see how the graph of a trig function (put the x in brackets e.g. sin(x)) relates to the differential of this function

scroll to user function
change the scales of the graph by entering values into the boxes then pressing 'graph'
use the slider to see the gradient function

differentiating trig functions

differentiating natural logarithms and exponentials

differentiating w.r.t. other variables

differentiating an inverse function

tangent and normal

finding the equation of the tangent and normal at a point involves substituting the x-coordinate into the gradient function (to find 'm') then using the x and y coordinates of the given point to work out what 'c' is

for this function, the turning point is identified (which can be related to the graph) and then the equations of the tangent and normal are found at the point where x = 9:

stationary points

quotient rule examples

differentiating with respect to different variables

use the chain rule to link derivatives (differentials) with respect to different variables

the product and quotient rule from first principles

product rule examples

product rule examples

(differentiate first) times (the second) then +  (the first) times (differential of the second)

when factorising, extract the lowest power of a common function

 

in this question, use the product rule to find the differential and then relate the turning points to the graph

some other examples, involving exponential, logarithmic and trig functions

product and quotient rule

 

an example of the product rule on Vimeo

an example of the quotient rule on Vimeo

chain rule

see this explained on Vimeo

here they use the terms 'inside function' and 'outside function'

the derivative is
the derivate of the outside function x
derivative of inside function

to see how the chain rule works, from some simple examples:

a couple of simple examples: