# Calculus Examples

, ,

Step 1

Step 1.1

Find .

Step 1.1.1

Differentiate both sides of the equation.

Step 1.1.2

The derivative of with respect to is .

Step 1.1.3

Differentiate the right side of the equation.

Step 1.1.3.1

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

Step 1.1.3.2

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

Step 1.1.3.2.1

To apply the Chain Rule, set as .

Step 1.1.3.2.2

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

Step 1.1.3.2.3

Replace all occurrences of with .

Step 1.1.3.3

Differentiate using the Power Rule which states that is where .

Step 1.1.3.4

Simplify.

Step 1.1.3.4.1

Reorder the factors of .

Step 1.1.3.4.2

Reorder factors in .

Step 1.1.4

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

Step 1.2

Substitute into the given differential equation.

Step 1.3

Reorder factors in .

Step 1.4

The given solution satisfies the given differential equation.

is a solution to

is a solution to

Step 2

Substitute in the initial condition.

Step 3

Step 3.1

Rewrite the equation as .

Step 3.2

Divide each term in by and simplify.

Step 3.2.1

Divide each term in by .

Step 3.2.2

Simplify the left side.

Step 3.2.2.1

Cancel the common factor of .

Step 3.2.2.1.1

Cancel the common factor.

Step 3.2.2.1.2

Divide by .

Step 3.2.3

Simplify the right side.

Step 3.2.3.1

Simplify the denominator.

Step 3.2.3.1.1

Raising to any positive power yields .

Step 3.2.3.1.2

Anything raised to is .

Step 3.2.3.2

Divide by .