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Calculus Examples
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Step 3.1
Differentiate using the Product Rule which states that is where and .
Step 3.2
Differentiate using the chain rule, which states that is where and .
Step 3.2.1
To apply the Chain Rule, set as .
Step 3.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Differentiate.
Step 3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.2
Combine and .
Step 3.3.3
Differentiate using the Power Rule which states that is where .
Step 3.3.4
Multiply by .
Step 3.3.5
By the Sum Rule, the derivative of with respect to is .
Step 3.3.6
Differentiate using the Power Rule which states that is where .
Step 3.3.7
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.8
Differentiate using the Power Rule which states that is where .
Step 3.3.9
Multiply by .
Step 3.3.10
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.11
Add and .
Step 3.4
Simplify.
Step 3.4.1
Apply the distributive property.
Step 3.4.2
Apply the distributive property.
Step 3.4.3
Combine terms.
Step 3.4.3.1
Combine and .
Step 3.4.3.2
Combine and .
Step 3.4.3.3
Multiply by .
Step 3.4.3.4
Combine and .
Step 3.4.3.5
Cancel the common factor of and .
Step 3.4.3.5.1
Factor out of .
Step 3.4.3.5.2
Cancel the common factors.
Step 3.4.3.5.2.1
Factor out of .
Step 3.4.3.5.2.2
Cancel the common factor.
Step 3.4.3.5.2.3
Rewrite the expression.
Step 3.4.3.5.2.4
Divide by .
Step 3.4.3.6
Combine and .
Step 3.4.3.7
Multiply by .
Step 3.4.3.8
Cancel the common factor of and .
Step 3.4.3.8.1
Factor out of .
Step 3.4.3.8.2
Cancel the common factors.
Step 3.4.3.8.2.1
Factor out of .
Step 3.4.3.8.2.2
Cancel the common factor.
Step 3.4.3.8.2.3
Rewrite the expression.
Step 3.4.3.8.2.4
Divide by .
Step 3.4.3.9
Move to the left of .
Step 3.4.3.10
Move .
Step 3.4.3.11
Add and .
Step 3.4.3.12
Subtract from .
Step 3.4.4
Reorder terms.
Step 3.4.5
Simplify each term.
Step 3.4.5.1
Simplify the numerator.
Step 3.4.5.1.1
Rewrite.
Step 3.4.5.1.2
Remove unnecessary parentheses.
Step 3.4.5.2
Move to the left of .
Step 3.4.6
Reorder factors in .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .