Implicit Differentiate, possibly?

Question

I am to find dy/dx for the following:

y = arctan (e^x^4)

Answer 1

Not implicit differentiation, you need the chain rule!

4x⁵ * e^z / (1 + e^z) where I have written z = x⁴ since this editor will not allow the exponential in the exponent.

4/6/2019

I have made a small but significant error.

If y = arctan u where u is a function of x, dy/dx = 1(1 + u²) (du/dx) ... this is the first application of the chain rule!

In this case u = e^z where z is defined as above and the derivative of this (again by the chain rule) is

4x⁵ * e^z.

The error I made was to forget that u is squared in the derivative of arctan...the final result then is

4x⁵*e^z/(1 + e^2z).

I hope that clarifies things!

Answer 2

Greetings! Lets solve this shall we!?

Lets find dy/dx implicitly!

Then,

y = arctan (e^x4) ..........................we take tangent to both sides

tany = e^x4 ....................................Then take the derivative to both sides

sec²y (dy/dx) = 4x³e^x4 ........................Then solve for (dy/dx)

dy/dx = (4x³e^x4) / sec²y.

Let sec²y = 1 + tan²y from our Trig Identities and replace it with tany = e^x4 such that

dy/dx = (4x³e^x4) / (1+ (tany)²)

dy/dx = (4x³e^x4) / (1 + (e^x4)²)

dy/dx = (4x³e^x4) / (1 + (e^x8))

I hope this helps!