# 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

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

Step 1.4

Differentiate using the Power Rule which states that is where .

Step 1.5

Add and .

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 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

Differentiate.

Step 8.2.1

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

Step 8.2.2

Since is constant with respect to , 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

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

Step 8.5

Simplify.

Step 8.5.1

Add and .

Step 8.5.2

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

Combine the opposite terms in .

Step 9.1.2.1

Subtract from .

Step 9.1.2.2

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 .