Question

Question #3a2d9

Answer 1

`dy/dx = (2x)/5 * 5((x^2+4)/5)^4 = 2x((x^2+4)/5)^4`

Explanation:

We can most easily do this via the Chain Rule. The Chain Rule states that, given a function composition `f(g(x)), (df)/dx = (dg)/dx *(df)/(dg)`. Considering the function `e^(x^2)`, we have `g(x) = x^2, f(h) = e^(g)`. Then `(dg)/dx = 2x, (df)/(dg) = e^(g)`, giving us `(df)/dx = 2x*e^(x^2)`

For this problem, we would have `y(x) = f(g(x)), g(x) = (x^2+4)/5, f(g) = g^5`. Then `(dg)/dx = (2x)/5, (df)/(dg) = 5g^4`. Substituting `(x^2+4)/5` back in for g, we get...

`dy/dx = (2x)/5 * 5((x^2+4)/5)^4 = 2x((x^2+4)/5)^4`