# 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.

Simplify each term.

Using the Binomial Theorem, expand to .

Apply the distributive property.

Simplify.

Multiply by .

Multiply by .

Remove parentheses.

The final answer is .

Find the components of the definition.

Plug in the components.

Simplify the numerator.

Apply the distributive property.

Simplify.

Multiply by .

Multiply by .

Subtract from .

Add and .

Subtract from .

Add and .

Subtract from .

Add and .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Raise to the power of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Divide by .

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.

Multiply .

Multiply by .

Multiply by .

Raising to any positive power yields .

Multiply by .

Add and .

Add and .

Simplify each term.

One to any power is one.

Multiply by .

Add and .

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 .

Multiply by .

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

Subtract from both sides of the equation.

Subtract from .

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