Let's start off with the function f(x)=(x²+7x)(x+4). We need to find the derivative, or the rate of change, of this function with respect to x.
We can see here that we are given a product of two functions, x²+7x and x+4. Hence, we will have to use the Product Rule of differentiation to find the derivative.
The Product Rule states that the derivative of a product of two functions is the derivative of the first times the second function plus the derivative of the second times the first function.
Let's follow this rule step by step:
1. Firstly, let's keep the first function as it is, i.e., x²+7x.
2. Then we take the derivative of the second function, x+4. The derivative is simply 1 as there are no exponents or other functions influencing x.
3. The product of the first function and the derivative of the second function is: (x²+7x)*1 = x²+7x.
Next,
4. we keep the second function (x+4) as it is
5. and compute the derivative of the first function (x²+7x). The derivative of x² is 2x, and the derivative of 7x is 7. So, the derivative of the first function is: 2x+7.
6. The product of the second function and the derivative of the first function is: (x+4)(2x+7).
Finally putting it all together according to the product rule, f'=(x²+7x)+ (x+4)(2x+7). This calculates the rate of change of our original function at any point x.
Thus, the derivative f' of the function f(x)=(x²+7x)(x+4) is given by f'(x)= x²+7x + (x+4)(2x+7).