How do you use the product Rule to find the derivative of #(10-2x)(6-2x)(x) + x^3#?

Question

Christopher Allers · Accepted Answer

#y^' = 15x^2 - 64x + 60#

Your function can be written as the sum of two functions, let's say #f(x)# and #g(x)#

#y = f(x) + g(x)#

where #f(x) = (10-2x)(6-2x) * x# and #g(x) = x^3#.

This means that the derivative of #y# will take the form

#d/dx(y) = d/dx(f(x)) + d/dx(g(x))#

Now, you can differentiate #f(x)#, which can be written as the product of three other functions, by using the product rule.

For a function #f(x)# that can be written as

#f(x) = h(x) * i(x) * k(x)#

you can find its derivative by using the formula

#color(blue)(d/dx(f(x)) = [d/dx(h(x))] * i(x) * k(x) + h(x) * [d/dx(i(x))] * k(x) + h(x) * i(x) * [d/dx(k(x))])#

In your case, you have

#d/dx(f(x)) = [d/dx(10-2x)] * (6-2x) * x + (10-2x) * [d/dx(6-2x)] * x + (10-2x) * (6-2x) * d/dx(x)#

#f^' = (-2) * (6-2x) * x + (10-2x) * (-2) * x + (10-2x) * (6-2x) * 1#

#f^' = -12x + 4x^2 -20x + 4x^2 + 60 - 32x + 4x^2#

#f^' = 12x^2 -64x + 60#

Your target derivative will thus be

#y^' = 12x^2 - 64x + 60 + d/dx(3x^2)#

#y^' = 12x^2 - 64x + 60 + 3x^2#

#y^' = color(green)(15x^2 - 64x + 60)#

Samantha Adkison · Answer

undefined To find the derivative using the product rule, differentiate each term separately and then sum them up. The product rule states that if you have two functions u(x) and v(x), the derivative of their product is u'(x)v(x) + u(x)v'(x). Apply this rule to each term in the expression (10-2x)(6-2x)(x) + x^3, then simplify the result.