Find the derivative of f(x) = (1 + 5x2)(x − x2) in two ways. use the product rule. f '(x) = perform the multiplication first. f '(x) = do your answers agree? yes no

The product rule states that given a function:
f(x)=g(x)h(x)
f'(x)=g'(x)h(x)+h'(x)g(x)
thus the derivative of the expression will be given by:
f(x) = (1 + 5x2)(x − x2)
g(x)=(1+5x^2)
g'(x)=10x

h(x)=(x-x^2)
h'(x)=1-2x
thus:
f'(x)=10x(x-x^2)+(1-2x)(1+5x^2)
simplifying this we get:
f'(x)=-20x^3+15x^2-12x+1

Answer Link

joaobezerra joaobezerra · Answer 2 · 2019-12-11T14:58:26+01:00

Answer:

The answers agree.

[tex]f^{\prime}(x) = - 20x^{3} + 15x^{2} - 2x + 1[/tex]

Step-by-step explanation:

The product rule is:

[tex]f(x) = g(x)h(x)[/tex]

[tex]f^{\prime}(x) = g^{\prime}(x)h(x) + g(x)h^{\prime}(x)[/tex]

In this problem, we have that:

[tex]f(x) = (1 + 5x^{2})(x - x^{2})[/tex]

So

[tex]g(x) = 1 + 5x^{2}[/tex]

[tex]g^{\prime}(x) = 10x[/tex]

[tex]h(x) = x - x^{2}[/tex]

[tex]h^{\prime}(x) = 1 - 2x[/tex].

[tex]f^{\prime}(x) = g^{\prime}(x)h(x) + g(x)h^{\prime}(x)[/tex]

[tex]f^{\prime}(x) = (10x)*(x - x^{2}) + (1 + 5x^{2})(1-2x)[/tex]

[tex]f^{\prime}(x) = 10x^{2} - 10x^{3} + 1 - 2x + 5x^{2} - 10x^{3}[/tex]

[tex]f^{\prime}(x) = - 20x^{3} + 15x^{2} - 2x + 1[/tex]

Multiplication first:

[tex]f(x) = (1 + 5x^{2})(x - x^{2})[/tex]

[tex]f(x) = x - x^{2} + 5x^{3} - 5x^{4}[/tex]

[tex]f(x) = -5x^{4} + 5x^{3} - x^{2} + x[/tex]

[tex]f^{\prime}(x) = -20x^{3} + 15x^{2} - 2x + 1[/tex]