Calculus Examples

Find the Tangent Line at the Point y=(7x)/(x-1) , (2,14)
,
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
Since is constant with respect to , the derivative of with respect to is .
Step 1.2
Differentiate using the Quotient Rule which states that is where and .
Step 1.3
Differentiate.
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Step 1.3.1
Differentiate using the Power Rule which states that is where .
Step 1.3.2
Multiply by .
Step 1.3.3
By the Sum Rule, the derivative of with respect to is .
Step 1.3.4
Differentiate using the Power Rule which states that is where .
Step 1.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.6
Simplify terms.
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Step 1.3.6.1
Add and .
Step 1.3.6.2
Multiply by .
Step 1.3.6.3
Subtract from .
Step 1.3.6.4
Simplify the expression.
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Step 1.3.6.4.1
Subtract from .
Step 1.3.6.4.2
Move the negative in front of the fraction.
Step 1.3.6.4.3
Multiply by .
Step 1.3.6.5
Combine and .
Step 1.3.6.6
Move the negative in front of the fraction.
Step 1.4
Evaluate the derivative at .
Step 1.5
Simplify.
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Step 1.5.1
Simplify the denominator.
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Step 1.5.1.1
Subtract from .
Step 1.5.1.2
One to any power is one.
Step 1.5.2
Simplify the expression.
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Step 1.5.2.1
Divide by .
Step 1.5.2.2
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
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
Add to both sides of the equation.
Step 2.3.2.2
Add and .
Step 3
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