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Calculus Examples
Step 1
Step 1.1
Differentiate with respect to .
Step 1.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.3
Differentiate using the Power Rule which states that is where .
Step 1.4
Multiply by .
Step 2
Step 2.1
Differentiate with respect to .
Step 2.2
By the Sum Rule, the derivative of with respect to is .
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 2.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.5
Add and .
Step 3
Step 3.1
Substitute for and for .
Step 3.2
Since the left side does not equal the right side, the equation is not an identity.
is not an identity.
is not an identity.
Step 4
Step 4.1
Substitute for .
Step 4.2
Substitute for .
Step 4.3
Substitute for .
Step 4.3.1
Substitute for .
Step 4.3.2
Subtract from .
Step 4.3.3
Substitute for .
Step 4.3.3.1
Cancel the common factor.
Step 4.3.3.2
Rewrite the expression.
Step 4.4
Find the integration factor .
Step 5
Step 5.1
The integral of with respect to is .
Step 5.2
Simplify the answer.
Step 5.2.1
Simplify.
Step 5.2.2
Exponentiation and log are inverse functions.
Step 6
Step 6.1
Multiply by .
Step 6.2
Multiply by by adding the exponents.
Step 6.2.1
Move .
Step 6.2.2
Multiply by .
Step 6.3
Multiply by .
Step 6.4
Apply the distributive property.
Step 6.5
Multiply by by adding the exponents.
Step 6.5.1
Multiply by .
Step 6.5.1.1
Raise to the power of .
Step 6.5.1.2
Use the power rule to combine exponents.
Step 6.5.2
Add and .
Step 7
Set equal to the integral of .
Step 8
Step 8.1
Since is constant with respect to , move out of the integral.
Step 8.2
By the Power Rule, the integral of with respect to is .
Step 8.3
Simplify the answer.
Step 8.3.1
Rewrite as .
Step 8.3.2
Simplify.
Step 8.3.2.1
Combine and .
Step 8.3.2.2
Combine and .
Step 8.3.3
Reorder terms.
Step 9
Since the integral of will contain an integration constant, we can replace with .
Step 10
Set .
Step 11
Step 11.1
Differentiate with respect to .
Step 11.2
By the Sum Rule, the derivative of with respect to is .
Step 11.3
Evaluate .
Step 11.3.1
Combine and .
Step 11.3.2
Combine and .
Step 11.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 11.3.4
Differentiate using the Power Rule which states that is where .
Step 11.3.5
Combine and .
Step 11.3.6
Combine and .
Step 11.3.7
Cancel the common factor of .
Step 11.3.7.1
Cancel the common factor.
Step 11.3.7.2
Divide by .
Step 11.4
Differentiate using the function rule which states that the derivative of is .
Step 11.5
Reorder terms.
Step 12
Step 12.1
Move all terms not containing to the right side of the equation.
Step 12.1.1
Subtract from both sides of the equation.
Step 12.1.2
Combine the opposite terms in .
Step 12.1.2.1
Subtract from .
Step 12.1.2.2
Add and .
Step 13
Step 13.1
Integrate both sides of .
Step 13.2
Evaluate .
Step 13.3
By the Power Rule, the integral of with respect to is .
Step 14
Substitute for in .
Step 15
Step 15.1
Combine and .
Step 15.2
Combine and .
Step 15.3
Combine and .