# Calculus Examples

Step 1

Step 1.1

Differentiate with respect to .

Step 1.2

By the Sum Rule, the derivative of with respect to is .

Step 1.3

Evaluate .

Step 1.3.1

Since is constant with respect to , the derivative of with respect to is .

Step 1.3.2

Differentiate using the Power Rule which states that is where .

Step 1.3.3

Multiply by .

Step 1.4

Differentiate using the Power Rule which states that is where .

Step 2

Step 2.1

Differentiate with respect to .

Step 2.2

Differentiate.

Step 2.2.1

By the Sum Rule, the derivative of with respect to is .

Step 2.2.2

Differentiate using the Power Rule which states that is where .

Step 2.3

Evaluate .

Step 2.3.1

Since is constant with respect to , the derivative of with respect to is .

Step 2.3.2

Differentiate using the Power Rule which states that is where .

Step 2.3.3

Multiply by .

Step 2.4

Differentiate using the Constant Rule.

Step 2.4.1

Since is constant with respect to , the derivative of with respect to is .

Step 2.4.2

Add and .

Step 3

Step 3.1

Substitute for and for .

Step 3.2

Since the two sides have been shown to be equivalent, the equation is an identity.

is an identity.

is an identity.

Step 4

Set equal to the integral of .

Step 5

Step 5.1

Split the single integral into multiple integrals.

Step 5.2

Since is constant with respect to , move out of the integral.

Step 5.3

By the Power Rule, the integral of with respect to is .

Step 5.4

Apply the constant rule.

Step 5.5

Combine and .

Step 5.6

Simplify.

Step 6

Since the integral of will contain an integration constant, we can replace with .

Step 7

Set .

Step 8

Step 8.1

Differentiate with respect to .

Step 8.2

By the Sum Rule, the derivative of with respect to is .

Step 8.3

Evaluate .

Step 8.3.1

Since is constant with respect to , the derivative of with respect to is .

Step 8.3.2

Differentiate using the Power Rule which states that is where .

Step 8.3.3

Multiply by .

Step 8.4

Evaluate .

Step 8.4.1

Since is constant with respect to , the derivative of with respect to is .

Step 8.4.2

Differentiate using the Power Rule which states that is where .

Step 8.4.3

Move to the left of .

Step 8.5

Differentiate using the function rule which states that the derivative of is .

Step 8.6

Reorder terms.

Step 9

Step 9.1

Move all terms not containing to the right side of the equation.

Step 9.1.1

Subtract from both sides of the equation.

Step 9.1.2

Subtract from both sides of the equation.

Step 9.1.3

Combine the opposite terms in .

Step 9.1.3.1

Subtract from .

Step 9.1.3.2

Add and .

Step 9.1.3.3

Subtract from .

Step 9.1.3.4

Add and .

Step 10

Step 10.1

Integrate both sides of .

Step 10.2

Evaluate .

Step 10.3

Apply the constant rule.

Step 11

Substitute for in .