Find the inverse of the function y=e^x+6x using derivatives.

We'll put f(x) = e^x+6x.

By
definition, f'(x)*[f^-1(x)]' = 1

[f^-1(x)]'=
1/f'(x)

f^-1(x) = Integral of
1/f'(x)

We'll calculate f'(x) = e^x +
6

[f^-1(x)]'= 1/(e^x +
6)

We'll calculate the indefinite
integral:

Int dx/(e^x +
6)

We'll put e^x + 6 = t => e^x = t -
6

We'll differentiate:

e^x*dx
= dt

dx = dt/e^x

dx =
dt/(t-6)

Int dx/(e^x + 6) = Int
dt/t*(t-6)

We'll decompose the fraction 1/t*(t-6) in a sum
or differenceof elementary fractions:

1/t*(t-6) = A/t +
B/(t-6)

1 = t(A+B) - 6A

A+B =
0

A = -B

A = -1/6 => B
= 1/6

1/t*(t-6) = -1/6t +
1/6(t-6)

Int dt/t*(t-6) = -Int dt/6t + Int
dt/6(t-6)

Int dt/t*(t-6) = (1/6)(Int dt/(t-6) - Int
dt/t)

Int dt/t*(t-6) = (1/6)(ln(t-6) - ln
t)

Int dt/t*(t-6) =
(1/6){ln[(t-6)/t]}

Int dx/(e^x + 6) = (1/6){ln[(e^x)/(e^x +
6)]} +C

The inverse function is f^-1(x) =
{ln[(e^x)/(e^x + 6)]}/6