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To take the derivative of y=(x^5+6x)^7
you will need the chain rule. The chain rule dictates you take the derivative of the outside function, multiply by the inside function, etc and work your way in.
This gets you:
`7(x^5 + 6x)^6 xx (x^5 + 6x) xx (5x^4 + 6)`
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.
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