Using a substitution to help differentiateWe will often need to differentiate functions that are more complex than the ones that we can already do. They will simply be variations, where ' x' has been replaced by a function ' f(x)'. Example: Differentiate the following: y = (3x − 2)4y = sin 5xy = 2e(2x + 1)y = cos 4x In each of these cases we can use a substitution to turn the expression into something we can differentiate. Answer to 1: We know how to differentiate x4, so we use the substitution u = (3x − 2) to turn the function into something that we can differentiate. This gives: y = (3x − 2)4Let u = 3x − 2 to give us, y = u4, Now differentiate to get: The only problem is that we want dy/dx, not dy/du, and this is where we use the chain rule. The chain rule says thatSo all we need to do is to multiply dy/du by du/dx. As u = 3x − 2, du/dx = 3, soAnswer to 2: Differentiate y = sin 5x. Let u = 5x (therefore, y = sin u)so using the chain ruleSo when using the chain rule: Express the original function as a simpler function of u, where u is a function of x. Differentiate the two functions you now have. Multiply the derivatives together, leaving your answer in terms of the original question (i. e. in x' s rather than u' s). For 3 and 4, see if you can do the workings and then check your answers against these: Answer to 3: Answer to 4: Using the chain rule to differentiate by inspectionWhen familiar with the chain rule, it is possible to produce a correct answer instantly without having to write down all the substitution working; simply follow through the three steps together. Example: Differentiate ln(x2 + 3x + 3)The denominator is from dy/du = 1/u, the numerator is du/dx)In each of these formulae we have used the substitution u = f(x) and so the f ′ (x) corresponds to(Have a go at using the chain rule to make the rules yourself.)