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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
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.2
Differentiate using the chain rule, which states that is where and .
Step 1.2.2.1
To apply the Chain Rule, set as .
Step 1.2.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.2.3
Replace all occurrences of with .
Step 1.2.3
Differentiate.
Step 1.2.3.1
Multiply by .
Step 1.2.3.2
By the Sum Rule, the derivative of with respect to is .
Step 1.2.3.3
Differentiate using the Power Rule which states that is where .
Step 1.2.4
Differentiate using the chain rule, which states that is where and .
Step 1.2.4.1
To apply the Chain Rule, set as .
Step 1.2.4.2
Differentiate using the Power Rule which states that is where .
Step 1.2.4.3
Replace all occurrences of with .
Step 1.2.5
Rewrite as .
Step 1.2.6
Simplify.
Step 1.2.6.1
Apply the distributive property.
Step 1.2.6.2
Reorder the factors of .
Step 1.3
Differentiate the right side of the equation.
Step 1.3.1
Differentiate.
Step 1.3.1.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.1.2
By the Sum Rule, the derivative of with respect to is .
Step 1.3.1.3
Differentiate using the Power Rule which states that is where .
Step 1.3.1.4
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 Power Rule which states that is where .
Step 1.3.2.3
Replace all occurrences of with .
Step 1.3.3
Multiply by .
Step 1.3.4
Rewrite as .
Step 1.3.5
Simplify.
Step 1.3.5.1
Apply the distributive property.
Step 1.3.5.2
Combine terms.
Step 1.3.5.2.1
Multiply by .
Step 1.3.5.2.2
Multiply by .
Step 1.3.5.3
Reorder terms.
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
Simplify .
Step 1.5.1.1
Rewrite.
Step 1.5.1.2
Simplify by adding zeros.
Step 1.5.1.3
Expand using the FOIL Method.
Step 1.5.1.3.1
Apply the distributive property.
Step 1.5.1.3.2
Apply the distributive property.
Step 1.5.1.3.3
Apply the distributive property.
Step 1.5.1.4
Simplify each term.
Step 1.5.1.4.1
Rewrite using the commutative property of multiplication.
Step 1.5.1.4.2
Multiply by by adding the exponents.
Step 1.5.1.4.2.1
Move .
Step 1.5.1.4.2.2
Multiply by .
Step 1.5.1.4.2.2.1
Raise to the power of .
Step 1.5.1.4.2.2.2
Use the power rule to combine exponents.
Step 1.5.1.4.2.3
Add and .
Step 1.5.1.4.3
Multiply by .
Step 1.5.1.4.4
Rewrite using the commutative property of multiplication.
Step 1.5.1.4.5
Multiply by .
Step 1.5.1.4.6
Multiply by .
Step 1.5.1.4.7
Multiply by by adding the exponents.
Step 1.5.1.4.7.1
Move .
Step 1.5.1.4.7.2
Multiply by .
Step 1.5.1.4.7.2.1
Raise to the power of .
Step 1.5.1.4.7.2.2
Use the power rule to combine exponents.
Step 1.5.1.4.7.3
Add and .
Step 1.5.1.4.8
Multiply by .
Step 1.5.2
Add to both sides of the equation.
Step 1.5.3
Move all terms not containing to the right side of the equation.
Step 1.5.3.1
Subtract from both sides of the equation.
Step 1.5.3.2
Subtract from both sides of the equation.
Step 1.5.4
Factor out of .
Step 1.5.4.1
Factor out of .
Step 1.5.4.2
Factor out of .
Step 1.5.4.3
Factor out of .
Step 1.5.4.4
Factor out of .
Step 1.5.4.5
Factor out of .
Step 1.5.5
Divide each term in by and simplify.
Step 1.5.5.1
Divide each term in by .
Step 1.5.5.2
Simplify the left side.
Step 1.5.5.2.1
Cancel the common factor of .
Step 1.5.5.2.1.1
Cancel the common factor.
Step 1.5.5.2.1.2
Rewrite the expression.
Step 1.5.5.2.2
Cancel the common factor of .
Step 1.5.5.2.2.1
Cancel the common factor.
Step 1.5.5.2.2.2
Rewrite the expression.
Step 1.5.5.2.3
Cancel the common factor of .
Step 1.5.5.2.3.1
Cancel the common factor.
Step 1.5.5.2.3.2
Divide by .
Step 1.5.5.3
Simplify the right side.
Step 1.5.5.3.1
Simplify each term.
Step 1.5.5.3.1.1
Cancel the common factor of and .
Step 1.5.5.3.1.1.1
Factor out of .
Step 1.5.5.3.1.1.2
Cancel the common factors.
Step 1.5.5.3.1.1.2.1
Factor out of .
Step 1.5.5.3.1.1.2.2
Cancel the common factor.
Step 1.5.5.3.1.1.2.3
Rewrite the expression.
Step 1.5.5.3.1.2
Cancel the common factor of and .
Step 1.5.5.3.1.2.1
Factor out of .
Step 1.5.5.3.1.2.2
Cancel the common factors.
Step 1.5.5.3.1.2.2.1
Factor out of .
Step 1.5.5.3.1.2.2.2
Cancel the common factor.
Step 1.5.5.3.1.2.2.3
Rewrite the expression.
Step 1.5.5.3.1.3
Move the negative in front of the fraction.
Step 1.5.5.3.1.4
Cancel the common factor of and .
Step 1.5.5.3.1.4.1
Factor out of .
Step 1.5.5.3.1.4.2
Cancel the common factors.
Step 1.5.5.3.1.4.2.1
Factor out of .
Step 1.5.5.3.1.4.2.2
Cancel the common factor.
Step 1.5.5.3.1.4.2.3
Rewrite the expression.
Step 1.5.5.3.1.5
Cancel the common factor of and .
Step 1.5.5.3.1.5.1
Factor out of .
Step 1.5.5.3.1.5.2
Cancel the common factors.
Step 1.5.5.3.1.5.2.1
Cancel the common factor.
Step 1.5.5.3.1.5.2.2
Rewrite the expression.
Step 1.5.5.3.1.6
Move the negative in front of the fraction.
Step 1.5.5.3.2
To write as a fraction with a common denominator, multiply by .
Step 1.5.5.3.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 1.5.5.3.3.1
Multiply by .
Step 1.5.5.3.3.2
Reorder the factors of .
Step 1.5.5.3.4
Combine the numerators over the common denominator.
Step 1.5.5.3.5
Combine the numerators over the common denominator.
Step 1.5.5.3.6
Multiply by by adding the exponents.
Step 1.5.5.3.6.1
Move .
Step 1.5.5.3.6.2
Multiply by .
Step 1.5.5.3.7
Factor out of .
Step 1.5.5.3.7.1
Factor out of .
Step 1.5.5.3.7.2
Factor out of .
Step 1.5.5.3.7.3
Factor out of .
Step 1.5.5.3.7.4
Factor out of .
Step 1.5.5.3.7.5
Factor out of .
Step 1.5.5.3.8
Factor out of .
Step 1.5.5.3.9
Rewrite as .
Step 1.5.5.3.10
Factor out of .
Step 1.5.5.3.11
Factor out of .
Step 1.5.5.3.12
Factor out of .
Step 1.5.5.3.13
Simplify the expression.
Step 1.5.5.3.13.1
Rewrite as .
Step 1.5.5.3.13.2
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
Simplify the numerator.
Step 1.7.3.1
Raise to the power of .
Step 1.7.3.2
Multiply by .
Step 1.7.3.3
One to any power is one.
Step 1.7.3.4
Multiply by .
Step 1.7.3.5
Subtract from .
Step 1.7.3.6
Add and .
Step 1.7.4
Multiply.
Step 1.7.4.1
Multiply by .
Step 1.7.4.2
Multiply by .
Step 1.7.5
Simplify the denominator.
Step 1.7.5.1
Raise to the power of .
Step 1.7.5.2
Multiply by .
Step 1.7.5.3
One to any power is one.
Step 1.7.5.4
Multiply by .
Step 1.7.5.5
Add and .
Step 1.7.5.6
Add and .
Step 1.7.6
Cancel the common factor of and .
Step 1.7.6.1
Factor out of .
Step 1.7.6.2
Cancel the common factors.
Step 1.7.6.2.1
Factor out of .
Step 1.7.6.2.2
Cancel the common factor.
Step 1.7.6.2.3
Rewrite the expression.
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
Combine and .
Step 2.3.1.5
Multiply .
Step 2.3.1.5.1
Multiply by .
Step 2.3.1.5.2
Combine and .
Step 2.3.1.5.3
Multiply by .
Step 2.3.1.6
Move to the left of .
Step 2.3.2
Move all terms not containing to the right side of the equation.
Step 2.3.2.1
Add to both sides of the equation.
Step 2.3.2.2
Write as a fraction with a common denominator.
Step 2.3.2.3
Combine the numerators over the common denominator.
Step 2.3.2.4
Add and .
Step 2.3.3
Write in form.
Step 2.3.3.1
Reorder terms.
Step 2.3.3.2
Remove parentheses.
Step 3