Question

y=(7x+5)^x

Answer 1

if you want to take the derivative, you could take the ln (natural log) of both sides, and then use implicit differentiation

or, you can rewrite 7x+5 as
\[ e^{(7x+5)} \]
and take the derivative of
\[ y= \left( e^{(7x+5)}\right)^x = e^{(7x+5)x}\]

Answer 2

I would suggest you use phi's first suggestion to differentiate implicitly: It is the simplest and most direct way to the answer.

1. ln(y) = x * ln (7x +5)

2. Differentiate using the ln differential rule to obtain:

(1/y) *y' = ( 1/(7x+5))* ( 7x+5)'

3. Use the polynomial differential rule on the last term:

(1/y) *y' = ( 1/(7x+5))* ( 7)

4. Re-organise terms:

y' = [ 7/(7x +5)] *y

5. You know what y is and can substitute it into the equation to obtain an expression in terms of x.