Calculus Examples

Find dy/dx y=(x^3+5)/((5x+3)(3x+5))
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Differentiate the right side of the equation.
Tap for more steps...
Step 3.1
Differentiate using the Quotient Rule which states that is where and .
Step 3.2
Differentiate.
Tap for more steps...
Step 3.2.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2.2
Differentiate using the Power Rule which states that is where .
Step 3.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.4
Add and .
Step 3.3
Differentiate using the Product Rule which states that is where and .
Step 3.4
Differentiate.
Tap for more steps...
Step 3.4.1
By the Sum Rule, the derivative of with respect to is .
Step 3.4.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.3
Differentiate using the Power Rule which states that is where .
Step 3.4.4
Multiply by .
Step 3.4.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.6
Simplify the expression.
Tap for more steps...
Step 3.4.6.1
Add and .
Step 3.4.6.2
Move to the left of .
Step 3.4.7
By the Sum Rule, the derivative of with respect to is .
Step 3.4.8
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.9
Differentiate using the Power Rule which states that is where .
Step 3.4.10
Multiply by .
Step 3.4.11
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.12
Simplify the expression.
Tap for more steps...
Step 3.4.12.1
Add and .
Step 3.4.12.2
Move to the left of .
Step 3.5
Simplify.
Tap for more steps...
Step 3.5.1
Apply the product rule to .
Step 3.5.2
Apply the distributive property.
Step 3.5.3
Apply the distributive property.
Step 3.5.4
Apply the distributive property.
Step 3.5.5
Simplify the numerator.
Tap for more steps...
Step 3.5.5.1
Simplify each term.
Tap for more steps...
Step 3.5.5.1.1
Rewrite using the commutative property of multiplication.
Step 3.5.5.1.2
Apply the distributive property.
Step 3.5.5.1.3
Multiply by .
Step 3.5.5.1.4
Multiply by .
Step 3.5.5.1.5
Expand using the FOIL Method.
Tap for more steps...
Step 3.5.5.1.5.1
Apply the distributive property.
Step 3.5.5.1.5.2
Apply the distributive property.
Step 3.5.5.1.5.3
Apply the distributive property.
Step 3.5.5.1.6
Simplify and combine like terms.
Tap for more steps...
Step 3.5.5.1.6.1
Simplify each term.
Tap for more steps...
Step 3.5.5.1.6.1.1
Rewrite using the commutative property of multiplication.
Step 3.5.5.1.6.1.2
Multiply by by adding the exponents.
Tap for more steps...
Step 3.5.5.1.6.1.2.1
Move .
Step 3.5.5.1.6.1.2.2
Multiply by .
Step 3.5.5.1.6.1.3
Multiply by .
Step 3.5.5.1.6.1.4
Multiply by .
Step 3.5.5.1.6.1.5
Multiply by .
Step 3.5.5.1.6.1.6
Multiply by .
Step 3.5.5.1.6.2
Add and .
Step 3.5.5.1.7
Apply the distributive property.
Step 3.5.5.1.8
Simplify.
Tap for more steps...
Step 3.5.5.1.8.1
Multiply by by adding the exponents.
Tap for more steps...
Step 3.5.5.1.8.1.1
Move .
Step 3.5.5.1.8.1.2
Use the power rule to combine exponents.
Step 3.5.5.1.8.1.3
Add and .
Step 3.5.5.1.8.2
Multiply by by adding the exponents.
Tap for more steps...
Step 3.5.5.1.8.2.1
Move .
Step 3.5.5.1.8.2.2
Multiply by .
Tap for more steps...
Step 3.5.5.1.8.2.2.1
Raise to the power of .
Step 3.5.5.1.8.2.2.2
Use the power rule to combine exponents.
Step 3.5.5.1.8.2.3
Add and .
Step 3.5.5.1.9
Multiply by .
Step 3.5.5.1.10
Simplify each term.
Tap for more steps...
Step 3.5.5.1.10.1
Multiply by .
Step 3.5.5.1.10.2
Multiply by .
Step 3.5.5.1.10.3
Multiply by .
Step 3.5.5.1.10.4
Multiply by .
Step 3.5.5.1.11
Add and .
Step 3.5.5.1.12
Add and .
Step 3.5.5.1.13
Expand using the FOIL Method.
Tap for more steps...
Step 3.5.5.1.13.1
Apply the distributive property.
Step 3.5.5.1.13.2
Apply the distributive property.
Step 3.5.5.1.13.3
Apply the distributive property.
Step 3.5.5.1.14
Simplify each term.
Tap for more steps...
Step 3.5.5.1.14.1
Rewrite using the commutative property of multiplication.
Step 3.5.5.1.14.2
Multiply by by adding the exponents.
Tap for more steps...
Step 3.5.5.1.14.2.1
Move .
Step 3.5.5.1.14.2.2
Multiply by .
Tap for more steps...
Step 3.5.5.1.14.2.2.1
Raise to the power of .
Step 3.5.5.1.14.2.2.2
Use the power rule to combine exponents.
Step 3.5.5.1.14.2.3
Add and .
Step 3.5.5.1.14.3
Multiply by .
Step 3.5.5.1.14.4
Multiply by .
Step 3.5.5.1.14.5
Multiply by .
Step 3.5.5.1.14.6
Multiply by .
Step 3.5.5.2
Subtract from .
Step 3.5.5.3
Subtract from .
Step 3.5.6
Reorder terms.
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .