Difference between $y=g(x^2)$ and $y=(g(x))^2$ when using chain rule

Question

The equation of the tangent line to the graph of $y=g(x)$ at 25 is: $$7x + 10y=65$$

1. What is the equation of the tangent line to $y=g(x^2)$ at $x=5$? $$y=?$$

2. What is the equation of the tangent line to $y=(g(x))^2$ at $x=25$? $$y=?$$

Answer 1

There are a lot of tutorials online about the chain rule, but to summarize, the difference between $g(x^2)$ and $g(x)^2$ is that you are composing two functions in a different order.

$g(x^2)$ applies $f(x)=x^2$ followed by $g(x)$ while $g(x)^2$ is the opposite.

When doing the chain rule, you would differentiate the 'outer' function first, followed by the inner one. So for 1, $y' = g'(x^2) \times 2x$ whereas for 2, $y' = 2g(x) \times g'(x)$

In the questions you gave, the 'tangent line' to the graph is basically $g'(x)$ at $25$, which should help with finding equations for 1 and 2.