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Calculus Examples
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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
Differentiate.
Step 1.2.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2.1.2
Differentiate using the Power Rule which states that is where .
Step 1.2.2
Evaluate .
Step 1.2.2.1
Differentiate using the Product Rule which states that is where and .
Step 1.2.2.2
Rewrite as .
Step 1.2.2.3
Differentiate using the Power Rule which states that is where .
Step 1.2.2.4
Multiply by .
Step 1.2.3
Evaluate .
Step 1.2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.3.2
Differentiate using the chain rule, which states that is where and .
Step 1.2.3.2.1
To apply the Chain Rule, set as .
Step 1.2.3.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3.2.3
Replace all occurrences of with .
Step 1.2.3.3
Rewrite as .
Step 1.2.3.4
Multiply by .
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
Move all terms not containing to the right side of the equation.
Step 1.5.1.1
Subtract from both sides of the equation.
Step 1.5.1.2
Subtract from both sides of the equation.
Step 1.5.2
Factor out of .
Step 1.5.2.1
Factor out of .
Step 1.5.2.2
Factor out of .
Step 1.5.2.3
Factor out of .
Step 1.5.3
Divide each term in by and simplify.
Step 1.5.3.1
Divide each term in by .
Step 1.5.3.2
Simplify the left side.
Step 1.5.3.2.1
Cancel the common factor of .
Step 1.5.3.2.1.1
Cancel the common factor.
Step 1.5.3.2.1.2
Divide by .
Step 1.5.3.3
Simplify the right side.
Step 1.5.3.3.1
Combine the numerators over the common denominator.
Step 1.5.3.3.2
Factor out of .
Step 1.5.3.3.3
Factor out of .
Step 1.5.3.3.4
Factor out of .
Step 1.5.3.3.5
Simplify the expression.
Step 1.5.3.3.5.1
Rewrite as .
Step 1.5.3.3.5.2
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
Reduce the expression by cancelling the common factors.
Step 1.7.3.1
Remove parentheses.
Step 1.7.3.2
Cancel the common factor of and .
Step 1.7.3.2.1
Rewrite as .
Step 1.7.3.2.2
Factor out of .
Step 1.7.3.2.3
Factor out of .
Step 1.7.3.2.4
Reorder terms.
Step 1.7.3.2.5
Factor out of .
Step 1.7.3.2.6
Factor out of .
Step 1.7.3.2.7
Factor out of .
Step 1.7.3.2.8
Cancel the common factors.
Step 1.7.3.2.8.1
Factor out of .
Step 1.7.3.2.8.2
Cancel the common factor.
Step 1.7.3.2.8.3
Rewrite the expression.
Step 1.7.3.3
Add and .
Step 1.7.4
Simplify the denominator.
Step 1.7.4.1
Multiply by .
Step 1.7.4.2
Subtract from .
Step 1.7.5
Simplify the expression.
Step 1.7.5.1
Multiply by .
Step 1.7.5.2
Move the negative in front of the fraction.
Step 1.7.6
Multiply .
Step 1.7.6.1
Multiply by .
Step 1.7.6.2
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
Combine and .
Step 2.3.1.5
Cancel the common factor of .
Step 2.3.1.5.1
Factor out of .
Step 2.3.1.5.2
Cancel the common factor.
Step 2.3.1.5.3
Rewrite the expression.
Step 2.3.1.6
Multiply by .
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
Add and .
Step 2.3.3
Reorder terms.
Step 3