Calculus Examples

Find the Tangent Line at (4,-1) f(x)=1/(x-5) ; (4,-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
Rewrite as .
Step 1.2
Differentiate using the chain rule, which states that is where and .
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Step 1.2.1
To apply the Chain Rule, set as .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Replace all occurrences of with .
Step 1.3
Differentiate.
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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
Simplify the expression.
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Step 1.3.4.1
Add and .
Step 1.3.4.2
Multiply by .
Step 1.4
Rewrite the expression using the negative exponent rule .
Step 1.5
Evaluate the derivative at .
Step 1.6
Simplify.
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Step 1.6.1
Simplify the denominator.
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Step 1.6.1.1
Subtract from .
Step 1.6.1.2
Raise to the power of .
Step 1.6.2
Reduce the expression by cancelling the common factors.
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Step 1.6.2.1
Cancel the common factor of .
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Step 1.6.2.1.1
Cancel the common factor.
Step 1.6.2.1.2
Rewrite the expression.
Step 1.6.2.2
Multiply 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
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
Simplify the expression.
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Step 2.3.1.4.1
Rewrite as .
Step 2.3.1.4.2
Multiply by .
Step 2.3.2
Move all terms not containing to the right side of the equation.
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Step 2.3.2.1
Subtract from both sides of the equation.
Step 2.3.2.2
Subtract from .
Step 3