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How do I differentiate y = (3 + 6x)^5 with respect to x

At first glance this seems as though we need to solve using the techniques of standard differentiation, however on further inspection we see we need to use a further method call the chain rule to solve this.

The chain rule uses the idea of dy/dx = dy/du  X  du/dx

(a way to remember to get the fractions the right way up on the right hand side, is to treat the entities as regular fractions and cancelling should leave dy/dx)

To use the chain rule substitute u = 3 + 6x

So y = u5

dy/du = 5u4 and du/dx = 6      (differentiate both equations)

sub these two differentials into dy/dx = dy/du  X  du/dx

So dy/dx = 5u4 X 6 = 30u4

Now substitute u = 3 + 6x back in

dy/dx = 30(3 + 6x)4

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