How do you use the product rule to differentiate #y = (x^2 + 2) (x^3 + 4)#?

Question

Emily Ammerman · Accepted Answer

If we consider the definition of product rule for the function #y = f(x)*g(x)#, we have #(dy)/(dx) = f'(x)g(x) + f(x)g'(x)#.

Applying such concept to your function, we have

#y = 2x(x^3+4)+(x^2+2)3x^2#

Simplifying:

#y = 2x^4+8x+3x^4+6x^2#

#y=5x^4+6x^2+8x#

Charles Anctil · Answer

undefined To use the product rule to differentiate $ y = (x^2 + 2)(x^3 + 4) $, follow these steps:

1. Identify the two functions being multiplied: $ f(x) = x^2 + 2 $ and $ g(x) = x^3 + 4 $.
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: $ f'(x) = 2x $ and $ g'(x) = 3x^2 $.
4. Use the product rule formula: $ (f(x)g(x))' = f'(x)g(x) + f(x)g'(x) $.
5. Substitute the derivatives and the original functions into the product rule formula: $ (x^2 + 2)(x^3 + 4)' = (2x)(x^3 + 4) + (x^2 + 2)(3x^2) $.
6. Simplify the expression: $ (x^2 + 2)(x^3 + 4)' = 2x^4 + 8x + 3x^4 + 6x^2 $.
7. Combine like terms: $ (x^2 + 2)(x^3 + 4)' = 5x^4 + 6x^2 + 8x $.

So, the derivative of $ y = (x^2 + 2)(x^3 + 4) $ is $ 5x^4 + 6x^2 + 8x $.