Question

How do you differentiate `f(x)=e^tan(1/x^2)` using the chain rule?

Answer 1

Use substitution

Explanation:

`f(x) = e^(tan(1/x^2))`
Assume `t=1/x^2` then `p = tan(t)` then
`f(p) = e^p`
`(df)/(dx) = (df)/(dp) * (dp)/(dt) *(dt)/(dx)`
`= e^p*sec^2t *(-2/x^3)`
Now back substitute

`= e^(tan(1/x^2)) \times sec^2(1/x^2) \times ((-2)/x^3)`

Another approach

Take `ln` on both sides
`ln(f(x)) = tan(1/x^2)`

Now differentiate both sides

`1/f \times (df)/(dx) = sec^2(1/x^2) \times ((-2)/x^3)`
`(df)/(dx) = f \times sec^2(1/x^2) \times ((-2)/x^3)`
`= e^(tan(1/x^2)) \times sec^2(1/x^2) \times ((-2)/x^3)`