# 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.

Step 1.1.3.3.1

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

Step 1.1.3.3.2

Differentiate using the Power Rule which states that is where .

Step 1.1.3.3.3

Simplify the expression.

Step 1.1.3.3.3.1

Multiply by .

Step 1.1.3.3.3.2

Move to the left of .

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

Remove parentheses.

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

Simplify .

Step 3.2.1

Multiply by .

Step 3.2.2

Anything raised to is .

Step 3.2.3

Multiply by .