How do you differentiate #g(y) = (x^2 - 1) (4x^6 + 5) # using the product rule?

Question

How do you differentiate #g(y) = (x^2 - 1) (4x^6 + 5)      # using the product rule?

Ezra Adams · Accepted Answer

#g'(x) = 2x(4x^6 + 5) + 24x^5(x^2 - 1)# #g# is the product of two functions #u# & #v# with #u(x) = x^2 - 1# & #v(x) = 4x^6 + 5# So the derivative of #g# is #u'v + uv'# with #u'(x) = 2x# & #v'(x) = 24x^5#.

Scarlett Allison · Answer

undefined To differentiate the function g(y) = (x^2 - 1) (4x^6 + 5) using the product rule, follow these steps:

1. Identify the two functions being multiplied: f(x) = x^2 - 1 and h(x) = 4x^6 + 5.
2. Apply the product rule, which states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.
3. Differentiate each function separately:
   - Differentiate f(x) with respect to x: f'(x) = 2x.
   - Differentiate h(x) with respect to x: h'(x) = 24x^5.
4. Apply the product rule formula: g'(x) = f(x)h'(x) + f'(x)h(x).
5. Substitute the expressions for f(x), h(x), f'(x), and h'(x) into the product rule formula.
6. Simplify the expression to obtain the derivative of g(x).

So, the derivative of g(y) = (x^2 - 1) (4x^6 + 5) using the product rule is:

g'(y) = (2x)(4x^6 + 5) + (x^2 - 1)(24x^5)
      = 8x^7 + 10x + 24x^7 - 24x^5
      = 32x^7 - 24x^5 + 10x.