# 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

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

Step 1.3.2

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

Step 1.3.2.1

To apply the Chain Rule, set as .

Step 1.3.2.2

Differentiate using the Exponential Rule which states that is where =.

Step 1.3.2.3

Replace all occurrences of with .

Step 1.3.3

Differentiate.

Step 1.3.3.1

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

Step 1.3.3.2

Multiply by .

Step 1.3.3.3

Differentiate using the Power Rule which states that is where .

Step 1.3.3.4

Multiply by .

Step 1.4

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

Step 2

Substitute into the given differential equation.

Step 3

Step 3.1

Multiply by .

Step 3.2

Add and .

Step 4

The given solution satisfies the given differential equation.

is a solution to