How do you differentiate #y=-x^2(-3x^2-2)# using the product rule?

Question

Elijah Abram · Accepted Answer

#12x^3 +4x# #y= x^2 (3x^2 +2)# #dy/dx= x^2 d/dx (3x^2 +2) +(3x^2 +2) d/dx (x^2)# =#x^2 (6x) +(3x^2 +2) (2x)# =#12x^3 +4x#

Christopher Allers · Answer

undefined To differentiate the given function $ y = -x^2(-3x^2 - 2) $ using the product rule, we follow these steps:

1. Identify the two functions being multiplied: $ u = -x^2 $ and $ v = -3x^2 - 2 $.
2. Apply the product rule formula $ (uv)' = u'v + uv' $.
3. Find the derivatives of $ u $ and $ v $: 
   - $ u' = -2x $ (derivative of $ -x^2 $).
   - $ v' = -6x $ (derivative of $ -3x^2 - 2 $).
4. Substitute the derivatives and the original functions into the product rule formula:
   - $ (uv)' = (-2x)(-3x^2 - 2) + (-x^2)(-6x) $.
5. Simplify the expression:
   - $ (-2x)(-3x^2 - 2) + (-x^2)(-6x) = 6x^3 + 4x + 6x^3 = 12x^3 + 4x $.

Therefore, the derivative of $ y = -x^2(-3x^2 - 2) $ with respect to $ x $ using the product rule is $ 12x^3 + 4x $.

Charles Arocha · Answer

undefined To differentiate y = -x²(-3x² - 2) using the product rule, first identify the two functions being multiplied: f(x) = -x² and g(x) = -3x² - 2. Then apply the product rule formula: (f(x)g(x))' = f'(x)g(x) + f(x)g'(x). Differentiate each function separately with respect to x, then apply the product rule formula to find the derivative of the product of the two functions.