Calculus Examples

Find the Derivative - d/dx y=(x^2-1)/(x^2+x+1)
Step 1
Differentiate using the Quotient Rule which states that is where and .
Step 2
Differentiate.
Tap for more steps...
Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.4
Simplify the expression.
Tap for more steps...
Step 2.4.1
Add and .
Step 2.4.2
Move to the left of .
Step 2.5
By the Sum Rule, the derivative of with respect to is .
Step 2.6
Differentiate using the Power Rule which states that is where .
Step 2.7
Differentiate using the Power Rule which states that is where .
Step 2.8
Since is constant with respect to , the derivative of with respect to is .
Step 2.9
Add and .
Step 3
Simplify.
Tap for more steps...
Step 3.1
Apply the distributive property.
Step 3.2
Apply the distributive property.
Step 3.3
Apply the distributive property.
Step 3.4
Simplify the numerator.
Tap for more steps...
Step 3.4.1
Simplify each term.
Tap for more steps...
Step 3.4.1.1
Multiply by by adding the exponents.
Tap for more steps...
Step 3.4.1.1.1
Move .
Step 3.4.1.1.2
Multiply by .
Tap for more steps...
Step 3.4.1.1.2.1
Raise to the power of .
Step 3.4.1.1.2.2
Use the power rule to combine exponents.
Step 3.4.1.1.3
Add and .
Step 3.4.1.2
Multiply by by adding the exponents.
Tap for more steps...
Step 3.4.1.2.1
Move .
Step 3.4.1.2.2
Multiply by .
Step 3.4.1.3
Multiply by .
Step 3.4.1.4
Multiply by .
Step 3.4.1.5
Expand using the FOIL Method.
Tap for more steps...
Step 3.4.1.5.1
Apply the distributive property.
Step 3.4.1.5.2
Apply the distributive property.
Step 3.4.1.5.3
Apply the distributive property.
Step 3.4.1.6
Simplify each term.
Tap for more steps...
Step 3.4.1.6.1
Rewrite using the commutative property of multiplication.
Step 3.4.1.6.2
Multiply by by adding the exponents.
Tap for more steps...
Step 3.4.1.6.2.1
Move .
Step 3.4.1.6.2.2
Multiply by .
Tap for more steps...
Step 3.4.1.6.2.2.1
Raise to the power of .
Step 3.4.1.6.2.2.2
Use the power rule to combine exponents.
Step 3.4.1.6.2.3
Add and .
Step 3.4.1.6.3
Multiply by .
Step 3.4.1.6.4
Multiply by .
Step 3.4.1.6.5
Multiply by .
Step 3.4.1.6.6
Multiply by .
Step 3.4.2
Combine the opposite terms in .
Tap for more steps...
Step 3.4.2.1
Subtract from .
Step 3.4.2.2
Add and .
Step 3.4.3
Subtract from .
Step 3.4.4
Add and .