Question

How do you find the derivative of `y=(e^(5x^4))/(e^(4x^2+3))`?

Answer 1

`dy/dx = (20x^3 - 8x)e^(5x^4 - 4x^2 - 3)`

Explanation:

Use the exponent law `x^a/x^n = x^(a - n)` to write as a single exponent.

`y= e^(5x^4 - (4x^2 + 3))`

`y = e^(5x^4 - 4x^2 - 3)`

Differentiate this using the chain rule. Let `y = e^u` and `u = 5x^4 - 4x^2 - 3`. We know that `d/dx(e^x) = e^x` and that `(du)/dx = 20x^3 - 8x`. Hence:

`dy/dx = dy/(du) * (du)/dx`

`dy/dx = e^u * 20x^3 - 8x`

`dy/dx = (20x^3 - 8x)e^(5x^4 - 4x^2 - 3)`

Hopefully this helps!