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

Question

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

William Abdalla · Accepted Answer

#2x(2x^2 +5)# The product rule states that if #h(x) = f(x)g(x)# then #h'(x)= f'(x)g(x) + f(x)g'(x)# #:. d/dxg(y) =( d/dx(x^2+6))(x^2-1) + (x^2+6)d/dx(x^2-1) # #= 2x(x^2-1) +(x^2+6)*2x# #=2x^3 -2x +2x^3 +12x# #=4x^3 +10x# #=2x(2x^2 +5)#

Evelyn Agard · Answer

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

1. Identify the functions $ f(x) = x^2 + 6 $ and $ h(x) = x^2 - 1 $.
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 individually:
   - $ f'(x) = 2x $
   - $ h'(x) = 2x $
4. Apply the product rule formula:
   - $ g'(x) = f'(x) \cdot h(x) + f(x) \cdot h'(x) $
5. Substitute the derivatives and original functions into the formula:
   - $ g'(x) = (2x)(x^2 - 1) + (x^2 + 6)(2x) $
6. Simplify the expression:
   - $ g'(x) = 2x^3 - 2x + 2x^3 + 12x $
   - $ g'(x) = 4x^3 + 10x $

So, the derivative of $ g(y) $ with respect to $ x $ using the product rule is $ 4x^3 + 10x $.