Calculus Examples

Find the Tangent Line at the Point y^4+x^3=y^2+10x , (0,1)
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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 chain rule, which states that is where and .
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Step 1.2.2.1.1
To apply the Chain Rule, set as .
Step 1.2.2.1.2
Differentiate using the Power Rule which states that is where .
Step 1.2.2.1.3
Replace all occurrences of with .
Step 1.2.2.2
Rewrite as .
Step 1.2.3
Differentiate using the Power Rule.
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Step 1.2.3.1
Differentiate using the Power Rule which states that is where .
Step 1.2.3.2
Reorder terms.
Step 1.3
Differentiate the right side of the equation.
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Step 1.3.1
By the Sum Rule, the derivative of with respect to is .
Step 1.3.2
Evaluate .
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Step 1.3.2.1
Differentiate using the chain rule, which states that is where and .
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Step 1.3.2.1.1
To apply the Chain Rule, set as .
Step 1.3.2.1.2
Differentiate using the Power Rule which states that is where .
Step 1.3.2.1.3
Replace all occurrences of with .
Step 1.3.2.2
Rewrite as .
Step 1.3.3
Evaluate .
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Step 1.3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3.3
Multiply by .
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
Subtract from both sides of the equation.
Step 1.5.3
Factor out of .
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Step 1.5.3.1
Factor out of .
Step 1.5.3.2
Factor out of .
Step 1.5.3.3
Factor out of .
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
Rewrite the expression.
Step 1.5.4.2.2
Cancel the common factor of .
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Step 1.5.4.2.2.1
Cancel the common factor.
Step 1.5.4.2.2.2
Rewrite the expression.
Step 1.5.4.2.3
Cancel the common factor of .
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Step 1.5.4.2.3.1
Cancel the common factor.
Step 1.5.4.2.3.2
Divide by .
Step 1.5.4.3
Simplify the right side.
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Step 1.5.4.3.1
Simplify each term.
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Step 1.5.4.3.1.1
Cancel the common factor of and .
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Step 1.5.4.3.1.1.1
Factor out of .
Step 1.5.4.3.1.1.2
Cancel the common factors.
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Step 1.5.4.3.1.1.2.1
Factor out of .
Step 1.5.4.3.1.1.2.2
Cancel the common factor.
Step 1.5.4.3.1.1.2.3
Rewrite the expression.
Step 1.5.4.3.1.2
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
Simplify each term.
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Step 1.7.3.1
Multiply by .
Step 1.7.3.2
Simplify the denominator.
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Step 1.7.3.2.1
One to any power is one.
Step 1.7.3.2.2
Multiply by .
Step 1.7.3.2.3
Subtract from .
Step 1.7.3.3
Divide by .
Step 1.7.3.4
Raising to any positive power yields .
Step 1.7.3.5
Multiply by .
Step 1.7.3.6
Multiply by .
Step 1.7.3.7
Simplify the denominator.
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Step 1.7.3.7.1
One to any power is one.
Step 1.7.3.7.2
Multiply by .
Step 1.7.3.7.3
Subtract from .
Step 1.7.3.8
Multiply by .
Step 1.7.3.9
Divide by .
Step 1.7.3.10
Multiply by .
Step 1.7.4
Add and .
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
Add and .
Step 2.3.2
Add to both sides of the equation.
Step 3