Finding the Slope of the Tangent Line to a Curve

In calculus, finding the slope of the tangent line to a curve defined implicitly can be a useful skill. Let's explore the process with a specific example:

The curve is defined implicitly by the equation:

Given Equation

8x^3 y - 15y^2 0

Implicit Differentiation

To find the slope of the tangent line to the curve, we need to implicitly differentiate the given equation with respect to x.

First, let's move everything to one side of the equation:

8x^3 y - 15y^2 0

Now, differentiate both sides with respect to x:

A Detailed Step-by-Step Differentiation

Step 1: Apply the Product Rule

For the term (8x^3 y), we use the product rule:

[ frac{d}{dx}(8x^3 y) 8 cdot 3x^2 y 8x^3 frac{dy}{dx} ]

Step 2: Differentiate (-15y^2)

The term (-15y^2) is differentiated with respect to y, and then multiplied by (frac{dy}{dx}):

[ frac{d}{dx}(-15y^2) -15 cdot 2y frac{dy}{dx} -30y frac{dy}{dx} ]

Putting these together:

[ 8 cdot 3x^2 y 8x^3 frac{dy}{dx} - 30y frac{dy}{dx} 0 ]

Combine like terms:

[ 24x^2 y (8x^3 - 30y) frac{dy}{dx} 0 ]

Solving for (frac{dy}{dx}):

[ (8x^3 - 30y) frac{dy}{dx} -24x^2 y ]

[ frac{dy}{dx} frac{-24x^2 y}{8x^3 - 30y} ]

Evaluating the Slope at a Specific Point

Now, let's evaluate the slope at the point ((-9, 5)).

Evaluating at ((-9, 5))

Substituting (x -9) and (y 5) into the derivative:

[ frac{dy}{dx} frac{-24(-9)^2 (5)}{8(-9)^3 - 30(5)} ]

[ frac{-24(81)(5)}{8(-729) - 150} ]

[ frac{-9720}{-5832 - 150} ]

[ frac{-9720}{-5982} ]

[ frac{9720}{5982} ]

[ frac{1620}{997} ]

[ approx -frac{57}{402} ]

Verification of the Point on the Curve

It is important to check that the point ((-9, 5)) lies on the curve. Substituting (x -9) and (y 5) into the original equation:

[ 8(-9)^3 (5) - 15(5)^2 0 ]

[ 8(-729)(5) - 15(25) 0 ]

[ -29160 - 375 0 ]

[ -29535 eq 0 ]

Therefore, the point ((-9, 5)) does not lie on the curve. This indicates a need to re-evaluate the point or the problem setup.

Conclusion

In summary, the slope of the tangent line to the curve at a specific point involves finding the derivative of the implicit function and evaluating it at the given point. It is crucial to verify that the point lies on the curve. If it does not, further investigation or correction of the point should be considered.