Lesson 54 — Implicit Differentiation

For the circle $x^2 + y^2 = 1$, find $dy/dx$.

For the ellipse $x^2/4 + y^2/9 = 1$, calculate $dy/dx$.

For $xy = 1$, calculate $dy/dx$ via implicit differentiation. Verify it matches differentiating $y = 1/x$ explicitly.

For the hyperbola $x^2/9 - y^2/16 = 1$, calculate $dy/dx$.

For $x^3 + y^3 = 6xy$, calculate $dy/dx$.

For $x^2 - 2xy + 3y^2 = 1$, calculate $dy/dx$.

For $x^2 y + xy^2 = 6$, calculate $dy/dx$.

For $\tan y = x$, calculate $dy/dx$. Interpret the result as the derivative of $\arctan x$.

For $e^y = xy$, calculate $dy/dx$.

For $\ln(xy) = x + y$, calculate $dy/dx$.

For $\sqrt{x} + \sqrt{y} = 4$, calculate $dy/dx$ and evaluate at the point $(1, 9)$.

For $\cos(x + y) = y$, calculate $dy/dx$.

For $\sin(xy) = x$, calculate $dy/dx$.

For $y^3 + 3y = x$, calculate $dy/dx$ and discuss whether the derivative exists at all points.

Find the tangent line to the circle $x^2 + y^2 = 25$ at the point $(3, 4)$.

For the ellipse $x^2 + 4y^2 = 16$, find the tangent line at the point $(2, \sqrt{3})$.

For $x^2 + xy + y^2 = 7$, find the tangent line at $(1, 2)$.

For $x^3 + y^3 = 9$, find the tangent line at $(1, 2)$.

For $y\sin x = x\cos y$, calculate $dy/dx$.

For the circle $x^2 + y^2 = 1$, determine all points of horizontal and vertical tangency.

For the Folium of Descartes $x^3 + y^3 = 3xy$, calculate $dy/dx$ and determine the points of horizontal tangency.

For the Folium of Descartes $x^3 + y^3 = 3xy$, find the tangent at the point $(3/2, 3/2)$.

The ideal gas law states $PV = nRT$. Holding $T$ constant, use implicit differentiation to find $dP/dV$.

For the curve $y^2 + xy = 12$, determine if there are any points of horizontal or vertical tangency.

In microeconomics, the indifference curve $U(x_1, x_2) = \bar{U}$ describes combinations of two goods that leave the consumer indifferent. Using implicit differentiation, find $dx_2/dx_1$ — the marginal rate of substitution.

For the lemniscate $(x^2+y^2)^2 = 2(x^2-y^2)$, calculate $dy/dx$ at the point $(\sqrt{3}/2, 1/2)$.

Use logarithmic differentiation to find $y'$ if $y = x^x$ ($x > 0$).

Use logarithmic differentiation to find $y'$ if $y = x^{\sin x}$ ($x > 0$). Evaluate at $x = \pi$.

For $x^2 + y^2 = r^2$, find $d^2y/dx^2$ in terms of $x$, $y$, and $r$. Interpret the sign of $y''$ for $y > 0$.

For the ellipse $x^2/a^2 + y^2/b^2 = 1$, calculate $dy/dx$ and $d^2y/dx^2$.

Why is the condition $F_y \neq 0$ necessary to apply the Implicit Function Theorem?

What is the main advantage of implicit differentiation over isolating $y$ and differentiating explicitly?

Use implicit differentiation to show that the tangent to the circle $x^2 + y^2 = r^2$ is always perpendicular to the radius at the point of tangency.

For a curve $F(x,y)=0$, explain the conditions under which the tangent line exists, possibly vertical, and when the point is singular.

Verify that differentiating $x^2 + y^2 = r^2$ implicitly gives the same result as differentiating $y = \pm\sqrt{r^2-x^2}$ explicitly.

When implicitly differentiating $e^{xy} = x + y$ with respect to $x$, what is $\frac{d}{dx}[e^y]$? Why is it not simply $e^y$?

For the curve $x^4 + y^4 = 1$, find all points of horizontal and vertical tangency.

For the ellipse $x^2/a^2 + y^2/b^2 = 1$, calculate $y''$ implicitly and simplify using the ellipse's equation. (Ans: $y'' = -b^4/(a^2 y^3)$.)

For $\sin(xy) + \cos(x+y) = 1$, calculate $dy/dx$ at $(0, 0)$. Explain why the point is singular for the direct formula.

Proof. Prove that $(x^a)' = ax^{a-1}$ for arbitrary $a \in \mathbb{R}$ ($x > 0$), using $x^a = e^{a\ln x}$ and the chain rule. Explain why the proof covers the case of irrational $a$.