Find the Tangent Line at the Point y=(4x)/(x-5) , (7,14)

Problem

y=(4*x)/(x−5),(7,14)

Solution

1. Identify the function ƒ(x)=(4*x)/(x−5) and the point ((x_0),(y_0))=(7,14)

2. Apply the quotient rule to find the derivative ƒ(x)′ which represents the slope of the tangent line.

ƒ(x)′=((x−5)(d(4)*x)/d(x)−(4*x)d(x−5)/d(x))/((x−5)2)

3. Differentiate the individual terms in the numerator.

ƒ(x)′=((x−5)*(4)−(4*x)*(1))/((x−5)2)

4. Simplify the expression for the derivative.

ƒ(x)′=(4*x−20−4*x)/((x−5)2)

ƒ(x)′=(−20)/((x−5)2)

5. Evaluate the derivative at x=7 to find the slope m

m=ƒ(7)′=(−20)/((7−5)2)

m=(−20)/2

m=(−20)/4

m=−5

6. Use the point-slope form y−(y_0)=m*(x−(x_0)) to write the equation of the tangent line.

y−14=−5*(x−7)

7. Simplify the equation into slope-intercept form y=m*x+b

y−14=−5*x+35

y=−5*x+49

Final Answer

y=−5*x+49

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