How do I differentiate y=(4+9x)^5 with respect to x?

The method we use to differentiate this form of equation is called the chain rule.

The chain rule is dy/dx = dy/du x du/dx

We can rememeber the right way up of the terms on the right hand side by treating them as fracions and cancelling to give dy/dx.

To use the chain rule we need to define our u. In this form of question we choose what is inside the brackets.

Let u=4+9x, this means that y=u^5.

Then by normal rules of differentiation we differentiate y and u giving:

dy/du = 5u^4   and    du/dx = 9

Then we substitue these results into the chain rule formula giving:

dy/dx = 9 x 5u^4 = 45u^4

Then we substitute u=4+9x back in to get our final answer:

dy/dx = 45(4+9x)^4

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