# Calculus Examples

Verify the Differential Equation Solution
,
Step 1
Find .
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
Simplify.
Step 3.1
Multiply by .
Step 3.2