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# Calculus Examples

,

Step 1

Step 1.1

Differentiate both sides of the equation.

Step 1.2

Differentiate the left side of the equation.

Step 1.2.1

By the Sum Rule, the derivative of with respect to is .

Step 1.2.2

Evaluate .

Step 1.2.2.1

Since is constant with respect to , the derivative of with respect to is .

Step 1.2.2.2

Differentiate using the Power Rule which states that is where .

Step 1.2.2.3

Multiply by .

Step 1.2.3

Evaluate .

Step 1.2.3.1

Differentiate using the chain rule, which states that is where and .

Step 1.2.3.1.1

To apply the Chain Rule, set as .

Step 1.2.3.1.2

Differentiate using the Power Rule which states that is where .

Step 1.2.3.1.3

Replace all occurrences of with .

Step 1.2.3.2

Rewrite as .

Step 1.2.4

Reorder terms.

Step 1.3

Since is constant with respect to , the derivative of with respect to is .

Step 1.4

Reform the equation by setting the left side equal to the right side.

Step 1.5

Solve for .

Step 1.5.1

Subtract from both sides of the equation.

Step 1.5.2

Divide each term in by and simplify.

Step 1.5.2.1

Divide each term in by .

Step 1.5.2.2

Simplify the left side.

Step 1.5.2.2.1

Cancel the common factor of .

Step 1.5.2.2.1.1

Cancel the common factor.

Step 1.5.2.2.1.2

Rewrite the expression.

Step 1.5.2.2.2

Cancel the common factor of .

Step 1.5.2.2.2.1

Cancel the common factor.

Step 1.5.2.2.2.2

Divide by .

Step 1.5.2.3

Simplify the right side.

Step 1.5.2.3.1

Cancel the common factor of and .

Step 1.5.2.3.1.1

Factor out of .

Step 1.5.2.3.1.2

Cancel the common factors.

Step 1.5.2.3.1.2.1

Factor out of .

Step 1.5.2.3.1.2.2

Cancel the common factor.

Step 1.5.2.3.1.2.3

Rewrite the expression.

Step 1.5.2.3.2

Move the negative in front of the fraction.

Step 1.6

Replace with .

Step 1.7

Evaluate at and .

Step 1.7.1

Replace the variable with in the expression.

Step 1.7.2

Replace the variable with in the expression.

Step 1.7.3

Move the negative one from the denominator of .

Step 1.7.4

Multiply .

Step 1.7.4.1

Multiply by .

Step 1.7.4.2

Multiply by .

Step 2

Step 2.1

Use the slope and a given point to substitute for and in the point-slope form , which is derived from the slope equation .

Step 2.2

Simplify the equation and keep it in point-slope form.

Step 2.3

Solve for .

Step 2.3.1

Simplify .

Step 2.3.1.1

Rewrite.

Step 2.3.1.2

Simplify by adding zeros.

Step 2.3.1.3

Apply the distributive property.

Step 2.3.1.4

Multiply by .

Step 2.3.2

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

Step 2.3.2.1

Subtract from both sides of the equation.

Step 2.3.2.2

Subtract from .

Step 3