How do you use the product Rule to find the derivative of #f(x) = (2x+1)(4-x^2)(1+x^2) #?

Question

Emily Ammerman · Accepted Answer

See the explanation.

#d/dx(gh) = g'h+gh'# #d/dx(uvw) = d/dx(u(vw))# # = u'(vw) + ud/dx(vw)# # = u'(vw) + u(v'w + vw')# # = u'vw + uv'w + uvw'# So for #f(x) = (2x+1)(4-x^2)(1+x^2)#, we get #f'(x) = (2)(4-x^2)(1+x^2)+(2x+1)(-2x)(1+x^2)+(2x+1)(4-x^2)(2x)#. Rewrite to taste using algebra.

Paisley Johnson · Answer

undefined To find the derivative of $ f(x) = (2x+1)(4-x^2)(1+x^2) $ using the product rule, follow these steps:

1. Identify the functions you'll apply the product rule to. In this case, you have three functions: $ u = 2x+1 $, $ v = 4-x^2 $, and $ w = 1+x^2 $.

2. Apply the product rule, which states that if $ f(x) = uvw $, then $ f'(x) = u'vw + uv'w + uvw' $.

3. Find the derivatives of $ u $, $ v $, and $ w $ with respect to $ x $: 
   $ u' = 2 $,
   $ v' = -2x $, and
   $ w' = 2x $.

4. Substitute the values of $ u $, $ v $, and $ w $, and their derivatives into the product rule formula.

5. Simplify the expression to get the derivative of $ f(x) $.

The derivative of $ f(x) $ is $ f'(x) = (2)(4-x^2)(1+x^2) + (2x+1)(-2x)(1+x^2) + (2x+1)(4-x^2)(2x) $.