# Calculus Examples

,

The slope of the tangent line is the derivative of the expression.

The derivative of

Consider the limit definition of the derivative.

Evaluate the function at .

Replace the variable with in the expression.

Simplify the result.

Simplify the expression.

Remove parentheses around .

Rewrite as .

Expand by multiplying each term in the first expression by each term in the second expression.

Simplify terms.

Remove unnecessary parentheses.

Simplify each term.

Use the power rule to combine exponents.

Add and to get .

Move to the left of the expression .

Multiply by to get .

Use the power rule to combine exponents.

Add and to get .

Move to the left of the expression .

Multiply by to get .

Multiply by to get .

Add and to get .

Reorder and .

Add and to get .

Add and to get .

Add and to get .

The final answer is .

Find the components of the definition.

Plug in the components.

Simplify the numerator.

Apply the distributive property.

Simplify.

Move parentheses.

Remove parentheses.

Subtract from to get .

Add and to get .

Subtract from to get .

Add and to get .

Subtract from to get .

Add and to get .

Factor out of .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by to get .

Split the limit using the Sum of Limits Rule on the limit as approaches .

Split the limit using the Product of Limits Rule on the limit as approaches .

Evaluate the limit of which is constant as approaches .

Evaluate the limit of which is constant as approaches .

Evaluate the limit of by plugging in for .

Evaluate the limit of which is constant as approaches .

Multiply by to get .

Add and to get .

Multiply by to get .

Add and to get .

The slope is and the point is .

Use the formula for the equation of a line to find .

Substitute the value of into the equation.

Substitute the value of into the equation.

Substitute the value of into the equation.

Find the value of .

Rewrite the equation as .

Simplify each term.

Multiply by to get .

Multiply by to get .

Move all terms not containing to the right side of the equation.

Subtract from both sides of the equation.

Add and to get .

Now that the values of (slope) and (y-intercept) are known, substitute them into to find the equation of the line.