Find the derivative of f(x) = (1 + 9x^2)(x − x^2) in two ways.Use the Product Rule and Perform the multiplication first.

Answer 1

For the product rule portion, I would first write down the two functions of x that are being multiplied.

ex;

f(x) = g(x) * h(x) & f'(x) = g'(x)h(x) + h'(x)g(x)

g(x) = 1+ 9x² and h(x) = x-x²

Identifying products in a derivative will be key to using this technique correctly as multiplying it out first is often time consuming or just isn't possible. At this point I would take the derivative of both functions and simply plug it into the product rule formula.

g'(x) = 18x and h'(x) = 1-2x with g(x) = 1+9x² and h(x) = x-x²

In this case, they're asking us to multiply it out, so our hands are tied.

after foiling you should find that f(x) = x-x^2 + 9x³ -9x⁴ and we can simply take the derivative term by term.

In either case, be sure to remember how exponents behave and the properties of exponents, specifically the case where x^a * x^b = x^(a+b)