How do I use the chain rule to differentiate polynomial powers of e?

e^{(x^2+2)}=f(x)=y

Is the equation we will use to demonstrate correct use of the chain rule.

The equation at the core of the chain rule is:

dy/dx=dt/dx*dy/dt

Seeing that dt as a numerator and dt as a denominator are both present in the equation allows us to cancel dt from the equation.

When using the chain rule, firstly, we must express f(x) using a simpler power of e, to do this we set t equal to x²+2, giving us the following equalities.

t=x²+2

y=e^t

From our differentiation rules we know that:

y=e^t

dy/dt=e^t

And:

t=x²+2

dt/dx=2x

Finally, we substitute into dy/dx=dt/dx*dy/dt 

(dy/dt)*(dt/dx)=dy/dx

(e^{(x^2+2)})*(2x)=dy/dx

y=e^{(x^2+2)}

dy/dx=2xe^{(x^2+2)}