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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 using the chain rule, which states that is where and .
Step 1.2.1.1
To apply the Chain Rule, set as .
Step 1.2.1.2
The derivative of with respect to is .
Step 1.2.1.3
Replace all occurrences of with .
Step 1.2.2
Differentiate using the Product Rule which states that is where and .
Step 1.2.3
Rewrite as .
Step 1.2.4
Differentiate using the Power Rule which states that is where .
Step 1.2.5
Multiply by .
Step 1.2.6
Simplify.
Step 1.2.6.1
Apply the distributive property.
Step 1.2.6.2
Reorder terms.
Step 1.3
Differentiate the right side of the equation.
Step 1.3.1
By the Sum Rule, the derivative of with respect to is .
Step 1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.4
Add and .
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 the left side.
Step 1.5.1.1
Reorder factors in .
Step 1.5.2
Subtract from both sides of the equation.
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
Rewrite the expression.
Step 1.5.3.2.2
Cancel the common factor of .
Step 1.5.3.2.2.1
Cancel the common factor.
Step 1.5.3.2.2.2
Divide by .
Step 1.5.3.3
Simplify the right side.
Step 1.5.3.3.1
Simplify each term.
Step 1.5.3.3.1.1
Cancel the common factor of .
Step 1.5.3.3.1.1.1
Cancel the common factor.
Step 1.5.3.3.1.1.2
Rewrite the expression.
Step 1.5.3.3.1.2
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 each term.
Step 1.7.3.1
Cancel the common factor of .
Step 1.7.3.1.1
Cancel the common factor.
Step 1.7.3.1.2
Rewrite the expression.
Step 1.7.3.2
Rewrite as .
Step 1.7.3.3
Rewrite as .
Step 1.7.3.4
Rewrite in terms of sines and cosines.
Step 1.7.3.5
Multiply by the reciprocal of the fraction to divide by .
Step 1.7.3.6
Simplify.
Step 1.7.3.6.1
Multiply by .
Step 1.7.3.6.2
Multiply by .
Step 1.7.3.7
The exact value of is .
Step 1.7.3.8
One to any power is one.
Step 1.7.3.9
Divide by .
Step 1.7.3.10
Multiply by .
Step 1.7.4
Add and .
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
Add and .
Step 2.3.2
Multiply by .
Step 3