Math Problem Statement
a. Use the product rule to find the derivative of the given function.
b. Find the derivative by expanding the product first.
h(z)equals=left parenthesis 6 minus z squared right parenthesis left parenthesis z cubed minus z plus 3 right parenthesis6−z2z3−z+3
Question content area bottom
Part 1
a. Use the product rule to find the derivative of the given
function.
Select the correct answer below and fill in the answer box(es) to complete your choice.
A.
The derivative is
left parenthesis 6 minus z squared right parenthesis left parenthesis z cubed minus z plus 3 right parenthesis plus left parenthesis nothing right parenthesis6−z2z3−z+3+enter your response here.
B.
The derivative is
left parenthesis 6 minus z squared right parenthesis left parenthesis nothing right parenthesis6−z2enter your response here.
C.
The derivative is
left parenthesis 6 minus z squared right parenthesis left parenthesis z cubed minus z plus 3 right parenthesis left parenthesis nothing right parenthesis6−z2z3−z+3enter your response here.
D.
The derivative is
left parenthesis 6 minus z squared right parenthesis left parenthesis nothing right parenthesis plus left parenthesis z cubed minus z plus 3 right parenthesis left parenthesis nothing right parenthesis6−z2enter your response here+z3−z+3enter your response here.
E.
The derivative is
left parenthesis z cubed minus z plus 3 right parenthesis left parenthesis nothing right parenthesisz3−z+3enter your response here.
Part 2
b. Expand the product.
left parenthesis 6 minus z squared right parenthesis left parenthesis z cubed minus z plus 3 right parenthesis6−z2z3−z+3equals=enter your response here
(Simplify your answer.)
Part 3
Using either approach,
StartFraction d Over dz EndFractionddzleft parenthesis 6 minus z squared right parenthesis left parenthesis z cubed minus z plus 3 right parenthesis6−z2z3−z+3equals=enter your response here.
Solution
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Math Problem Analysis
Mathematical Concepts
Differentiation
Product Rule
Polynomial Expansion
Formulas
Product Rule: d/dz [f(z) * g(z)] = f'(z) * g(z) + f(z) * g'(z)
Derivative of a Polynomial: d/dz [a_n z^n + ... + a_1 z + a_0] = n*a_n z^(n-1) + ... + a_1
Theorems
Product Rule of Differentiation
Suitable Grade Level
Grades 11-12, College Calculus