Find the Tangent Line at the Point y=(7x)/(x-3) , (6,14)

Problem

y=(7*x)/(x−3),(6,14)

Solution

1. Identify the function and the given point ((x_0),(y_0))=(6,14)

2. Apply the quotient rule to find the derivative d(y)/d(x) which represents the slope of the tangent line.

d(y)/d(x)=((x−3)⋅(d(7)*x)/d(x)−7*x⋅d(x−3)/d(x))/((x−3)2)

3. Differentiate the terms in the numerator.

d(y)/d(x)=((x−3)*(7)−7*x(1))/((x−3)2)

4. Simplify the derivative expression.

d(y)/d(x)=(7*x−21−7*x)/((x−3)2)

d(y)/d(x)=(−21)/((x−3)2)

5. Evaluate the derivative at x=6 to find the slope m

m=(−21)/((6−3)2)

m=(−21)/3

m=(−21)/9

m=−7/3

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

y−14=−7/3*(x−6)

7. Solve for y to put the equation in slope-intercept form.

y−14=−7/3*x+14

y=−7/3*x+28

Final Answer

y=−7/3*x+28

Want more problems? Check here!