Calculus Examples

Find the Derivative Using Product Rule - d/d@VAR f(z)=(9z^4+3z^2+1)(3z^3-z)
Step 1
Differentiate using the Product Rule which states that is where and .
Step 2
By the Sum Rule, the derivative of with respect to is .
Step 3
Evaluate .
Tap for more steps...
Step 3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2
Differentiate using the Power Rule which states that is where .
Step 3.3
Multiply by .
Step 4
Evaluate .
Tap for more steps...
Step 4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.2
Differentiate using the Power Rule which states that is where .
Step 4.3
Multiply by .
Step 5
By the Sum Rule, the derivative of with respect to is .
Step 6
Evaluate .
Tap for more steps...
Step 6.1
Since is constant with respect to , the derivative of with respect to is .
Step 6.2
Differentiate using the Power Rule which states that is where .
Step 6.3
Multiply by .
Step 7
Evaluate .
Tap for more steps...
Step 7.1
Since is constant with respect to , the derivative of with respect to is .
Step 7.2
Differentiate using the Power Rule which states that is where .
Step 7.3
Multiply by .
Step 8
Differentiate using the Constant Rule.
Tap for more steps...
Step 8.1
Since is constant with respect to , the derivative of with respect to is .
Step 8.2
Add and .
Step 9
Simplify.
Tap for more steps...
Step 9.1
Factor out of .
Tap for more steps...
Step 9.1.1
Factor out of .
Step 9.1.2
Factor out of .
Step 9.1.3
Factor out of .
Step 9.2
Multiply by .
Step 9.3
Reorder terms.
Step 9.4
Simplify each term.
Tap for more steps...
Step 9.4.1
Expand by multiplying each term in the first expression by each term in the second expression.
Step 9.4.2
Simplify each term.
Tap for more steps...
Step 9.4.2.1
Rewrite using the commutative property of multiplication.
Step 9.4.2.2
Multiply by by adding the exponents.
Tap for more steps...
Step 9.4.2.2.1
Move .
Step 9.4.2.2.2
Use the power rule to combine exponents.
Step 9.4.2.2.3
Add and .
Step 9.4.2.3
Multiply by .
Step 9.4.2.4
Rewrite using the commutative property of multiplication.
Step 9.4.2.5
Multiply by by adding the exponents.
Tap for more steps...
Step 9.4.2.5.1
Move .
Step 9.4.2.5.2
Use the power rule to combine exponents.
Step 9.4.2.5.3
Add and .
Step 9.4.2.6
Multiply by .
Step 9.4.2.7
Multiply by .
Step 9.4.2.8
Multiply by .
Step 9.4.2.9
Multiply by .
Step 9.4.2.10
Multiply by .
Step 9.4.3
Subtract from .
Step 9.4.4
Subtract from .
Step 9.4.5
Expand using the FOIL Method.
Tap for more steps...
Step 9.4.5.1
Apply the distributive property.
Step 9.4.5.2
Apply the distributive property.
Step 9.4.5.3
Apply the distributive property.
Step 9.4.6
Simplify and combine like terms.
Tap for more steps...
Step 9.4.6.1
Simplify each term.
Tap for more steps...
Step 9.4.6.1.1
Rewrite using the commutative property of multiplication.
Step 9.4.6.1.2
Multiply by by adding the exponents.
Tap for more steps...
Step 9.4.6.1.2.1
Move .
Step 9.4.6.1.2.2
Use the power rule to combine exponents.
Step 9.4.6.1.2.3
Add and .
Step 9.4.6.1.3
Multiply by .
Step 9.4.6.1.4
Rewrite using the commutative property of multiplication.
Step 9.4.6.1.5
Multiply by by adding the exponents.
Tap for more steps...
Step 9.4.6.1.5.1
Move .
Step 9.4.6.1.5.2
Multiply by .
Tap for more steps...
Step 9.4.6.1.5.2.1
Raise to the power of .
Step 9.4.6.1.5.2.2
Use the power rule to combine exponents.
Step 9.4.6.1.5.3
Add and .
Step 9.4.6.1.6
Multiply by .
Step 9.4.6.1.7
Rewrite using the commutative property of multiplication.
Step 9.4.6.1.8
Multiply by by adding the exponents.
Tap for more steps...
Step 9.4.6.1.8.1
Move .
Step 9.4.6.1.8.2
Multiply by .
Tap for more steps...
Step 9.4.6.1.8.2.1
Raise to the power of .
Step 9.4.6.1.8.2.2
Use the power rule to combine exponents.
Step 9.4.6.1.8.3
Add and .
Step 9.4.6.1.9
Multiply by .
Step 9.4.6.1.10
Rewrite using the commutative property of multiplication.
Step 9.4.6.1.11
Multiply by by adding the exponents.
Tap for more steps...
Step 9.4.6.1.11.1
Move .
Step 9.4.6.1.11.2
Multiply by .
Step 9.4.6.1.12
Multiply by .
Step 9.4.6.2
Add and .
Step 9.5
Combine the opposite terms in .
Tap for more steps...
Step 9.5.1
Subtract from .
Step 9.5.2
Add and .
Step 9.5.3
Subtract from .
Step 9.5.4
Add and .
Step 9.6
Add and .