# Calculus Examples

Verify the Differential Equation Solution
,
Step 1
Find .
Differentiate both sides of the equation.
The derivative of with respect to is .
Differentiate the right side of the equation.
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Exponential Rule which states that is where =.
Replace all occurrences of with .
Differentiate.
Since is constant with respect to , the derivative of with respect to is .
Multiply by .
Differentiate using the Power Rule which states that is where .
Multiply by .
Reform the equation by setting the left side equal to the right side.
Step 2
Substitute into the given differential equation.
Step 3
Simplify.
Multiply by .