How do you differentiate using the chain rule?

At first the chain rule can seem quite daunting and convoluted but with a few examples over the course of this lesson, it will seem simpler and more intuitive.
The chain rule is used where the equation that one is looking to differentiate is a function that is itself raised to a power. For example, y = (3x - 2)^4 which needs to be differentiating with respect to x to give dy/dx.
What methods can be applied in order to answer this question?We could multiply this function out to give a full equation, but this can be messy, especially if the outside power (4 in the above example) is highThe alternative (and superior) method is to use the chain rule in order to work out the answer
NB. - when using the chain rule, a substitution is used to turn the expression into something that can be differentiated. For example, y= (3x-2)^4We already know (assumed here if discussing chain rule) how to differentiate x^4, so we use the substitution u = (3x-2) to turn the function into something that can be differentiated. This gives: y = (3x-2)^4Let u = (3x-2)to give us: y = u^4 dy/du = 4u^3
The only problem is that we want dy/dx, not dy/du and this is where we see the chain rule.The chain rule says that: dy/dx = dy/du x du/dx
So all we need to do is multiply dy/du by du/dx As u = 3x-2, du/dx = 3, sody/dx = 4u^3 x 3 = 12u^3 = 12(3x-2)^3
So, in summary, when using the chain rule:Express the original function as a simpler function of u, where u is a function of xDifferentiate the two functions you now haveMultiply the derivatives together, leaving your answer in terms of the original question (i.e. - in x's rather than u's)
Further practice:Differentiate with respect to x the following:y = sin(5x)y = 2e^(2x+1)