What rule is useful in finding derivative of the function y=(x^5+6x)^7 ?

We'll have to use the chain rule since the given
function is the result of composition of 2 functions.

u(x)
= x^5+6x and v(u) = u^7

y = f(x) = (vou)(x) = v(u(x)) =
v(x^5+6x) = (x^5+6x)^7

We'll differentiate f(x) and we'll
get:

f'(x) =
v'(u(x))*u'(x)

First, we'll differentiate v with respect to
u:

v'(u) = 7u^(7-1) =
7u^6

Second, we'll differentiate u with respect to
x:

u'(x) = (x^5+6x)' = 5x^4 +
6

f'(x) = 7u^6*(5x^4 +
6)

We'll substitute u and we'll get the derivative of f(x)
= y.

The derivative of f(x) is: f'(x) =
7*(5x^4 + 6)*(x^5+6x)^6.