Implicit differentiation and inverse trigonometric functions, question 21

Topic: Implicit differentiation and inverse trigonometric functions
Difficulty: Challenging
Suggested time: 6 min
Marks: 4
Theory: UBC Math 100 §4.5 CLP-1 §2.11Key results

Verify that the point $(2, 4)$ lies on the curve below, then find the equation of the tangent line there.

x^{3} + y^{3} = 9xy

Give the tangent line in any correct form, then compare against the model solution.

Your answer

Theory UBC Math 100 §4.5 CLP-1 §2.11Key results

Worked solution

Strategy

Confirm the point is on the curve, differentiate implicitly to get a slope formula in x and y, evaluate it at the point, and assemble the point-slope form.

Working

$2^{3} + 4^{3} = 8 + 64 = 72, \qquad 9(2)(4) = 72$
Both sides agree, so the point does lie on the curve.
$3x^{2} + 3y^{2}\frac{dy}{dx} = 9y + 9x\frac{dy}{dx}$
Differentiate both sides. The right-hand side is a product, so it splits into two terms.
$\frac{dy}{dx}\bigl(3y^{2} - 9x\bigr) = 9y - 3x^{2}$
Move every dy/dx term to the left and everything else to the right.
$\frac{dy}{dx} = \frac{9y - 3x^{2}}{3y^{2} - 9x} = \frac{3y - x^{2}}{y^{2} - 3x}$
Divide, then cancel the common factor of 3.
$\left.\frac{dy}{dx}\right|_{(2,\,4)} = \frac{3(4) - 4}{16 - 6} = \frac{8}{10} = \frac{4}{5}$
Substitute the point to get the numerical slope.
$y - 4 = \frac{4}{5}(x - 2)$
Point-slope form. Equivalently y = (4/5)x + 12/5.

Answer

y - 4 = \frac{4}{5}(x - 2)

What earns the marks

Checks the point satisfies the equation: 8 + 64 = 72 and 9(2)(4) = 72.
Differentiates implicitly, applying the product rule to the right-hand side.
Solves for dy/dx and substitutes both coordinates to get a slope of 4/5.
States a correct equation for the tangent line, such as y − 4 = (4/5)(x − 2).

Check

The curve also passes through about (2.01, 4.0080), a rise of about 0.008 over a step of 0.01, a slope of about 0.8 = 4/5.

Where this goes wrong

Differentiating 9xy as 9·dy/dx, treating x as a constant. It is a genuine product, so it differentiates to 9y + 9x·dy/dx.

Leans on

Equation of a tangent line, from The definition of the derivative

Try a similar question

The point

(1, 2)

lies on the curve below. Find

\dfrac{dy}{dx}

at that point.

x^{2} + xy + y^{2} = 7

My working

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Implicit differentiation and inverse trigonometric functions practice question 21