Find the derivative of the function.

f(x) = ex8 - 6

To find the derivative of the function f(x) = e^x^8 - 6, we can use the chain rule, since we have a composition of functions.

The chain rule states that if we have a function g(x) = f(h(x)), where f is a function of one variable and h is another function, then the derivative of g(x) with respect to x is given by g'(x) = f'(h(x)) * h'(x).

In our case, let's call h(x) = x^8 and f(x) = e^x.

Now, let's find the derivatives of f(x) and h(x):

The derivative of f(x) = e^x with respect to x is simply f'(x) = e^x.

The derivative of h(x) = x^8 with respect to x can be found using the power rule, which states that if we have a function g(x) = x^n, then the derivative of g(x) with respect to x is given by g'(x) = n * x^(n-1).

Applying the power rule, we have h'(x) = 8 * x^(8-1) = 8x^7.

Now, we can apply the chain rule to find the derivative of f(x) = e^x^8 with respect to x:

f'(x) = f'(h(x)) * h'(x) = e^(x^8) * 8x^7.

Therefore, the derivative of the function f(x) = e^x^8 - 6 is f'(x) = e^(x^8) * 8x^7.