Calculus Examples

Find the Tangent Line at the Point y = natural log of x^2-8 , (3,0)
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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 using the chain rule, which states that is where and .
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Step 1.1.1
To apply the Chain Rule, set as .
Step 1.1.2
The derivative of with respect to is .
Step 1.1.3
Replace all occurrences of with .
Step 1.2
Differentiate.
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Step 1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.4
Combine fractions.
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Step 1.2.4.1
Add and .
Step 1.2.4.2
Combine and .
Step 1.2.4.3
Combine and .
Step 1.3
Evaluate the derivative at .
Step 1.4
Simplify.
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Step 1.4.1
Multiply by .
Step 1.4.2
Simplify the denominator.
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Step 1.4.2.1
Raise to the power of .
Step 1.4.2.2
Subtract from .
Step 1.4.3
Divide by .
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
Simplify .
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Step 2.3.2.1
Apply the distributive property.
Step 2.3.2.2
Multiply by .
Step 3