Calculus Examples

Find the Tangent Line at the Point x^(2/3)+y^(2/3)=4 , (-3 square root of 3,1)
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Step 1
Find the first derivative and evaluate at and to find the slope of the tangent line.
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Step 1.1
Differentiate both sides of the equation.
Step 1.2
Differentiate the left side of the equation.
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Step 1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2.2
Evaluate .
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Step 1.2.2.1
Differentiate using the Power Rule which states that is where .
Step 1.2.2.2
To write as a fraction with a common denominator, multiply by .
Step 1.2.2.3
Combine and .
Step 1.2.2.4
Combine the numerators over the common denominator.
Step 1.2.2.5
Simplify the numerator.
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Step 1.2.2.5.1
Multiply by .
Step 1.2.2.5.2
Subtract from .
Step 1.2.2.6
Move the negative in front of the fraction.
Step 1.2.3
Evaluate .
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Step 1.2.3.1
Differentiate using the chain rule, which states that is where and .
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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 1.2.3.3
To write as a fraction with a common denominator, multiply by .
Step 1.2.3.4
Combine and .
Step 1.2.3.5
Combine the numerators over the common denominator.
Step 1.2.3.6
Simplify the numerator.
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Step 1.2.3.6.1
Multiply by .
Step 1.2.3.6.2
Subtract from .
Step 1.2.3.7
Move the negative in front of the fraction.
Step 1.2.3.8
Combine and .
Step 1.2.3.9
Combine and .
Step 1.2.3.10
Move to the denominator using the negative exponent rule .
Step 1.2.4
Simplify.
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Step 1.2.4.1
Rewrite the expression using the negative exponent rule .
Step 1.2.4.2
Multiply by .
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 .
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Step 1.5.1
Subtract from both sides of the equation.
Step 1.5.2
Multiply both sides by .
Step 1.5.3
Simplify.
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Step 1.5.3.1
Simplify the left side.
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Step 1.5.3.1.1
Simplify .
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Step 1.5.3.1.1.1
Rewrite using the commutative property of multiplication.
Step 1.5.3.1.1.2
Cancel the common factor of .
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Step 1.5.3.1.1.2.1
Cancel the common factor.
Step 1.5.3.1.1.2.2
Rewrite the expression.
Step 1.5.3.1.1.3
Cancel the common factor of .
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Step 1.5.3.1.1.3.1
Cancel the common factor.
Step 1.5.3.1.1.3.2
Rewrite the expression.
Step 1.5.3.2
Simplify the right side.
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Step 1.5.3.2.1
Simplify .
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Step 1.5.3.2.1.1
Cancel the common factor of .
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Step 1.5.3.2.1.1.1
Move the leading negative in into the numerator.
Step 1.5.3.2.1.1.2
Factor out of .
Step 1.5.3.2.1.1.3
Factor out of .
Step 1.5.3.2.1.1.4
Cancel the common factor.
Step 1.5.3.2.1.1.5
Rewrite the expression.
Step 1.5.3.2.1.2
Combine and .
Step 1.5.3.2.1.3
Move the negative in front of the fraction.
Step 1.5.4
Divide each term in by and simplify.
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Step 1.5.4.1
Divide each term in by .
Step 1.5.4.2
Simplify the left side.
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Step 1.5.4.2.1
Cancel the common factor of .
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Step 1.5.4.2.1.1
Cancel the common factor.
Step 1.5.4.2.1.2
Divide by .
Step 1.5.4.3
Simplify the right side.
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Step 1.5.4.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 1.5.4.3.2
Cancel the common factor of .
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Step 1.5.4.3.2.1
Move the leading negative in into the numerator.
Step 1.5.4.3.2.2
Factor out of .
Step 1.5.4.3.2.3
Cancel the common factor.
Step 1.5.4.3.2.4
Rewrite the expression.
Step 1.5.4.3.3
Move the negative in front of the fraction.
Step 1.6
Replace with .
Step 1.7
Evaluate at and .
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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
One to any power is one.
Step 1.7.4
Apply the product rule to .
Step 2
Plug the slope and point values into the point-slope formula and solve for .
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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 .
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Step 2.3.1
Simplify .
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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 .
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Step 2.3.1.5.1
Multiply by .
Step 2.3.1.5.2
Combine and .
Step 2.3.1.5.3
Combine and .
Step 2.3.1.6
Simplify each term.
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Step 2.3.1.6.1
Move to the numerator using the negative exponent rule .
Step 2.3.1.6.2
Move to the numerator using the negative exponent rule .
Step 2.3.1.6.3
Multiply by by adding the exponents.
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Step 2.3.1.6.3.1
Move .
Step 2.3.1.6.3.2
Multiply by .
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Step 2.3.1.6.3.2.1
Raise to the power of .
Step 2.3.1.6.3.2.2
Use the power rule to combine exponents.
Step 2.3.1.6.3.3
Write as a fraction with a common denominator.
Step 2.3.1.6.3.4
Combine the numerators over the common denominator.
Step 2.3.1.6.3.5
Add and .
Step 2.3.1.6.4
Multiply by by adding the exponents.
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Step 2.3.1.6.4.1
Move .
Step 2.3.1.6.4.2
Multiply by .
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Step 2.3.1.6.4.2.1
Raise to the power of .
Step 2.3.1.6.4.2.2
Use the power rule to combine exponents.
Step 2.3.1.6.4.3
Write as a fraction with a common denominator.
Step 2.3.1.6.4.4
Combine the numerators over the common denominator.
Step 2.3.1.6.4.5
Add and .
Step 2.3.1.6.5
Rewrite as .
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Step 2.3.1.6.5.1
Use to rewrite as .
Step 2.3.1.6.5.2
Apply the power rule and multiply exponents, .
Step 2.3.1.6.5.3
Multiply by .
Step 2.3.1.6.5.4
Multiply by .
Step 2.3.1.6.5.5
Cancel the common factor of and .
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Step 2.3.1.6.5.5.1
Factor out of .
Step 2.3.1.6.5.5.2
Cancel the common factors.
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Step 2.3.1.6.5.5.2.1
Factor out of .
Step 2.3.1.6.5.5.2.2
Cancel the common factor.
Step 2.3.1.6.5.5.2.3
Rewrite the expression.
Step 2.3.1.6.5.6
Rewrite as .
Step 2.3.2
Add to both sides of the equation.
Step 2.3.3
Write in form.
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Step 2.3.3.1
Reorder terms.
Step 2.3.3.2
Remove parentheses.
Step 3