derivatives of log and exponents?

Question

r(x)=(e^41x^2)^3

t(x)=3^7x-3

r(x)=e^6x^2-8x+1/x

Answer 1

Since these functions are composite functions of more than one function, you need to apply the chain rule, which states:
d/dx f(g(x) = f'(g(x) * g'(x)

For example r(x) is the composition of f(x) = x^3 and g(x) = e^41x^2
(notice that g(x) is itself a composite function)
Thus, r'(x) =
3 (e^41x^2)^2 * d/dx (e^41x^2) =
3 (e^41x^2)^2 * (e^41x^2) *d/dx (41x^2) =
3 (e^41x^2)^2 * (e^41x^2) * 82x

t'(x) =
3^(7x-3) * ln3 * d/dx (7x-3) =
3^(7x-3) * ln3 * 7x

Assuming the last problem is e^(6x^2 - 8x + 1/x)
r'(x) =
e^(6x^2 - 8x + 1/x) * d/dx (6x^2 - 8x + 1/x) =
e^(6x^2 - 8x + 1/x) * (12x - 8 -1/x^2)

Answer 2

r´(x) = e^123x^2*(246x)
t´(x) =7*3^(7x-3) *ln3
r´(x) = e^(6x^2-8x+1/x)*(12x-8-1/x^2)