Calculus Examples

Find the Derivative - d/dx y = eighth root of (x+2)/(x+3)
Step 1
Use to rewrite as .
Step 2
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 2.1
To apply the Chain Rule, set as .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
Replace all occurrences of with .
Step 3
To write as a fraction with a common denominator, multiply by .
Step 4
Combine and .
Step 5
Combine the numerators over the common denominator.
Step 6
Simplify the numerator.
Tap for more steps...
Step 6.1
Multiply by .
Step 6.2
Subtract from .
Step 7
Move the negative in front of the fraction.
Step 8
Differentiate using the Quotient Rule which states that is where and .
Step 9
Differentiate.
Tap for more steps...
Step 9.1
By the Sum Rule, the derivative of with respect to is .
Step 9.2
Differentiate using the Power Rule which states that is where .
Step 9.3
Since is constant with respect to , the derivative of with respect to is .
Step 9.4
Simplify the expression.
Tap for more steps...
Step 9.4.1
Add and .
Step 9.4.2
Multiply by .
Step 9.5
By the Sum Rule, the derivative of with respect to is .
Step 9.6
Differentiate using the Power Rule which states that is where .
Step 9.7
Since is constant with respect to , the derivative of with respect to is .
Step 9.8
Combine fractions.
Tap for more steps...
Step 9.8.1
Add and .
Step 9.8.2
Multiply by .
Step 9.8.3
Multiply by .
Step 9.8.4
Move to the left of .
Step 10
Simplify.
Tap for more steps...
Step 10.1
Change the sign of the exponent by rewriting the base as its reciprocal.
Step 10.2
Apply the product rule to .
Step 10.3
Apply the distributive property.
Step 10.4
Combine terms.
Tap for more steps...
Step 10.4.1
Multiply by .
Step 10.4.2
Subtract from .
Step 10.4.3
Add and .
Step 10.4.4
Subtract from .
Step 10.4.5
Multiply by .
Step 10.4.6
Move to the denominator using the negative exponent rule .
Step 10.4.7
Multiply by by adding the exponents.
Tap for more steps...
Step 10.4.7.1
Move .
Step 10.4.7.2
Use the power rule to combine exponents.
Step 10.4.7.3
To write as a fraction with a common denominator, multiply by .
Step 10.4.7.4
Combine and .
Step 10.4.7.5
Combine the numerators over the common denominator.
Step 10.4.7.6
Simplify the numerator.
Tap for more steps...
Step 10.4.7.6.1
Multiply by .
Step 10.4.7.6.2
Add and .