Finding the derivative- is my answer correct? Is the method correct?

Question

Question - find the derivative of $y$ with respect to $z$ $$y= 2^{z^3} $$

So what I’d do is Differentiate boths sides of the equation so $$\frac{dy}{dz}= \frac{d(2^{z^3 })}{dz}$$ Then I applied the chain rule by assuming, $$ u=z^3$$

which resulted in $$ 2^{z^3}\ln2\frac{dz^3}{dz}$$

Using the power rule I finally got the answer to be $$ 3z^2\cdot2^{z^3}\ln2$$

Is my answer correct?

Answer 1

The chain rule in this case gives: $$\frac{dy}{dz}=\frac{dy}{du}\cdot\frac{du}{dz}$$

You set $u=z^3$, so $y=2^u$, then $\frac{du}{dz}=3z^2$ and $\frac{dy}{du}=2^u\cdot \ln (2)$

Combining the two and substituting out $u$ we get $$\frac{du}{dz}\cdot\frac{dy}{du}=\frac{dy}{dz}=3\ln (2) \cdot 2^{z^3}\cdot z^2$$ which you got.