Calculus Examples

Find the Tangent Line at the Point y=(7x)/(x+4) , (3,3)
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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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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Quotient Rule which states that is where and .
Differentiate.
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Differentiate using the Power Rule which states that is where .
Multiply by .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Simplify terms.
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Add and .
Multiply by .
Subtract from .
Add and .
Combine and .
Multiply by .
Evaluate the derivative at .
Simplify.
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Simplify the denominator.
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Add and .
Raise to the power of .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Step 2
Plug the slope and point values into the point-slope formula and solve for .
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Use the slope and a given point to substitute for and in the point-slope form , which is derived from the slope equation .
Simplify the equation and keep it in point-slope form.
Solve for .
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Simplify .
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Rewrite.
Simplify by adding zeros.
Apply the distributive property.
Combine and .
Multiply .
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Combine and .
Multiply by .
Move the negative in front of the fraction.
Move all terms not containing to the right side of the equation.
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Add to both sides of the equation.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Add and .
Reorder terms.
Step 3
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