How to differentiate using the chain rule

The chain rule is a rule of differentiation that states: (f(g(x))'=f^'(g(x))g'(x)

In effect, the chain rule allows us to split up a complicated equation into smaller more manageable chunks. Rather than differentiating f(g(x)) all in one go, all we need to do is differentiate f(.) - giving us f^'(g(x)) -  and then differentiate g(.) - giving us g'(x). We can then multiply these together to find the derrivative of the original equation.

Consider the following example:

e^3x+8=y

At first this equation may seem complicated and difficult to differentiate, in fact, it's quite easy. All we need to do is apply the chain rule.

In this case, we can see that the equation is made up of two "chunks" which we can seperate out.

Firstly there is the simple linear equation:

g(x)=3x+8

and then there is also the exponential function:

e^g(x)=f(g(x))

Then it is easy to see that f(g(x))=y (just substitute g(x)=3x+8 into the equation e^g(x))  This is an equation of exactly the form we need in order to use the chain rule.

Now we have a much simpler problem, we just need to differentiate each "chunk"

g'(x)= 3     (this is just a linear equation so this easy to check)

f^'(g(x))=e^g(x)   (Since the derrivative of the exponential function is simply the very same function).

We can then multiply these together to find the derrivative of the original equation:

dy/dx = 3e^3x+8