Sure. Let's take this step by step.
We have a function f(x) = 5e^(3x^7 + 8x^3).
The first step in derivating this function is to recognise that it's composed of a function inside another function - a composition of functions - making this a prime candidate for applying the chain rule.
The chain rule states that the derivative of a composite function is the derivative of the outside function times the derivative of the inside function.
First, we need to identify our 'inside' and 'outside' functions. Here, the outside function g(x) is 5e^x and the inside function h(x) is (3x^7 + 8x^3).
Step 1: Compute the derivative of g(x) = 5e^x
The derivative of e^x is just e^x. So, the derivative of 5e^x would be 5e^x.
Step 2: Compute the derivative of h(x) = 3x^7 + 8x^3
For the term 3x^7, we bring down the exponent 7 and reduce the exponent by 1 to get 21x^6. Similarly, for the term 8x^3, we bring down the exponent 3 and reduce it by 1 to get 24x^2.
Step 3: Apply the Chain Rule
According to the chain rule, the derivative of our original function f(x) is the product of the derivative of g(x) and the derivative of h(x). So derivative f'(x) would be 5e^(3x^7 + 8x^3) multiplied with (21x^6 + 24x^2).
So the derivative of the function f(x)=5e^(3x^7 + 8x^3) is
f'(x) =5*(21x^6 + 24x^2)*e^(3x^7 + 8x^3).