# 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.

Remove parentheses.

Remove unnecessary parentheses.

Simplify each term.

Using the Binomial Theorem, expand to .

Apply the distributive property.

Simplify.

Multiply by to get .

Multiply by to get .

Simplify each term.

Remove parentheses around .

Remove parentheses around .

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 .

Move the exponent from outside the limit using the Limits Power Rule.

Move the term outside of the limit because it is constant with respect to .

Move the term outside of the limit because it is constant with respect to .

Move the exponent from outside the limit using the Limits Power Rule.

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 by plugging in for .

Evaluate the limit of which is constant as approaches .

Simplify each term.

Remove parentheses around .

Multiply by to get .

Simplify .

Multiply by to get .

Multiply by to get .

Remove parentheses around .

Raising to any positive power yields .

Multiply by to get .

Add and to get .

Add and to get .

Simplify each term.

Remove parentheses around .

One to any power is one.

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.

Since does not contain the variable to solve for, move it to the right side of the equation by subtracting from both sides.

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.