# Calculus Examples

,

Step 1

Step 1.1

Differentiate both sides of the equation.

Step 1.2

The derivative of with respect to is .

Step 1.3

Differentiate the right side of the equation.

Step 1.3.1

Differentiate using the chain rule, which states that is where and .

Step 1.3.1.1

To apply the Chain Rule, set as .

Step 1.3.1.2

The derivative of with respect to is .

Step 1.3.1.3

Replace all occurrences of with .

Step 1.3.2

Differentiate.

Step 1.3.2.1

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

Step 1.3.2.2

Differentiate using the Power Rule which states that is where .

Step 1.3.2.3

Simplify the expression.

Step 1.3.2.3.1

Multiply by .

Step 1.3.2.3.2

Reorder the factors of .

Step 1.4

Reform the equation by setting the left side equal to the right side.

Step 2

Step 2.1

Set up the derivative.

Step 2.2

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

Step 2.3

Differentiate using the chain rule, which states that is where and .

Step 2.3.1

To apply the Chain Rule, set as .

Step 2.3.2

The derivative of with respect to is .

Step 2.3.3

Replace all occurrences of with .

Step 2.4

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

Step 2.5

Raise to the power of .

Step 2.6

Raise to the power of .

Step 2.7

Use the power rule to combine exponents.

Step 2.8

Add and .

Step 2.9

Differentiate using the Power Rule which states that is where .

Step 2.10

Multiply by .

Step 2.11

Reorder the factors of .

Step 3

Substitute into the given differential equation.

Step 4

Substitute for .

Step 5

Step 5.1

Multiply by .

Step 5.2

Subtract from both sides of the equation.

Step 5.3

Divide each term in by and simplify.

Step 5.3.1

Divide each term in by .

Step 5.3.2

Simplify the left side.

Step 5.3.2.1

Cancel the common factor of .

Step 5.3.2.1.1

Cancel the common factor.

Step 5.3.2.1.2

Rewrite the expression.

Step 5.3.2.2

Cancel the common factor of .

Step 5.3.2.2.1

Cancel the common factor.

Step 5.3.2.2.2

Divide by .

Step 5.3.3

Simplify the right side.

Step 5.3.3.1

Cancel the common factor of .

Step 5.3.3.1.1

Cancel the common factor.

Step 5.3.3.1.2

Rewrite the expression.

Step 5.3.3.2

Dividing two negative values results in a positive value.

Step 5.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

Step 5.5

Simplify .

Step 5.5.1

Rewrite as .

Step 5.5.2

Any root of is .

Step 5.5.3

Multiply by .

Step 5.5.4

Combine and simplify the denominator.

Step 5.5.4.1

Multiply by .

Step 5.5.4.2

Raise to the power of .

Step 5.5.4.3

Raise to the power of .

Step 5.5.4.4

Use the power rule to combine exponents.

Step 5.5.4.5

Add and .

Step 5.5.4.6

Rewrite as .

Step 5.5.4.6.1

Use to rewrite as .

Step 5.5.4.6.2

Apply the power rule and multiply exponents, .

Step 5.5.4.6.3

Combine and .

Step 5.5.4.6.4

Cancel the common factor of .

Step 5.5.4.6.4.1

Cancel the common factor.

Step 5.5.4.6.4.2

Rewrite the expression.

Step 5.5.4.6.5

Evaluate the exponent.

Step 5.6

The complete solution is the result of both the positive and negative portions of the solution.

Step 5.6.1

First, use the positive value of the to find the first solution.

Step 5.6.2

Next, use the negative value of the to find the second solution.

Step 5.6.3

The complete solution is the result of both the positive and negative portions of the solution.

Step 6

The result can be shown in multiple forms.

Exact Form:

Decimal Form: