Implicit Differentiation
Comprehensive Worksheet

Introduction

There is a quiet revolution that happens somewhere in every calculus course — the moment a student realizes that not every curve on the coordinate plane can be described by a neat formula of the form $y = f(x)$. A circle, for instance. Or a folium. Or the graph of $\sin(xy) = x^2 - y$. These curves exist. They have slopes at their points. Yet no amount of algebraic cleverness will isolate $y$ on one side of the equation in a usable closed form. The question, then, is not whether to differentiate these relations. It is how.

Implicit differentiation is the answer. Its intellectual roots trace back to Leibniz himself, who in the late seventeenth century understood that the relationship between two variables could be far richer than a simple functional assignment. When Newton and Leibniz built the calculus, they envisioned it as a tool for understanding rates of change in general — not just for functions in the modern sense. Implicit differentiation is, in a way, the most faithful expression of that original vision.

The technique rests on a single, elegant idea: we treat $y$ as a function of $x$ — even if we cannot write out what that function is explicitly — and we differentiate both sides of an equation with respect to $x$. The chain rule then requires that every derivative involving $y$ carries a factor of $dy/dx$, which is precisely the quantity we want to find. After differentiating both sides, we collect all $dy/dx$ terms on one side and solve algebraically. The result is a formula for $dy/dx$ that typically contains both $x$ and $y$ — not a defect, but a feature. The slope at any point on the curve can be found by substituting the coordinates of that point.

The technique is not merely a computational trick. It is a profound extension of the very concept of a derivative. In multivariable calculus, implicit differentiation evolves into the implicit function theorem — one of the cornerstones of advanced analysis. In differential equations, it underlies the method of exact equations. In physics, it is used whenever two dynamical quantities are constrained by a physical law and both are changing in time (the machinery known as related rates is implicit differentiation in disguise). The curve in a phase diagram. The constraint surface in a thermodynamic system. The level set in an optimization problem. All of these are implicitly defined relations, and all of them can be interrogated using the techniques you will practice here.

This worksheet is designed for students who have already encountered basic differentiation rules — the power rule, product and quotient rules, and the chain rule — and are now ready to see those tools applied in a setting that demands greater structural awareness. If you can differentiate $f(x) = \sin(x^3)$ and correctly account for the chain rule, you are prepared to begin. If you can also differentiate $f(x) = x^2 e^{\sin x}$ without hesitation, you will find most of this worksheet deeply satisfying rather than overwhelming.

Mastery of implicit differentiation looks like this: you encounter a relation between $x$ and $y$, your eye immediately identifies which terms require the chain rule and which require the product rule, you differentiate systematically without losing sign or coefficient, and you isolate $dy/dx$ with algebraic confidence. You can find slopes, write tangent line equations, classify horizontal and vertical tangencies, and compute second derivatives. That is the full picture, and this worksheet builds toward it in stages.

Conceptual Foundation

Formal Definition and Core Procedure

The formal justification comes from the chain rule applied to composite functions. If $y = y(x)$ is differentiable, then for any differentiable function $g$:

This is the chain rule with outer function $g$ and inner function $y(x)$. Implicit differentiation is simply the disciplined application of this rule to every $y$-containing term in an equation, combined with the linearity of differentiation (which means we can differentiate an equation term-by-term).

When to Use Implicit Differentiation

Structural Pattern Recognition

Before applying the procedure, scan the equation for its structural features. Terms that are purely in $x$ (like $x^3$ or $\cos x$) differentiate normally. Terms purely in $y$ (like $y^4$ or $e^y$) produce a $dy/dx$ factor. The real complexity — and the most common source of errors — comes from mixed terms like $x^2 y$, $xy^3$, or $\frac{x}{y}$, which require the product or quotient rule on top of the chain rule.

The Second Derivative

Finding $d^2y/dx^2$ implicitly requires differentiating the expression for $dy/dx$ again with respect to $x$. Since $dy/dx$ typically contains both $x$ and $y$, this second differentiation will itself require implicit differentiation — and will generate another $dy/dx$ term, into which you substitute the first derivative expression. The algebra intensifies, but the logic does not.

Comparison to Related Techniques

Students sometimes confuse implicit differentiation with logarithmic differentiation. The techniques overlap when the equation can be made implicit by taking a logarithm (useful for expressions like $y = x^x$), but they are distinct strategies. Logarithmic differentiation is typically chosen to simplify products and powers; implicit differentiation is chosen because $y$ cannot be isolated. For the chain rule, the connection is even closer: implicit differentiation simply is the chain rule, applied systematically across an entire equation rather than to a single composite expression.

Worked Examples

Practice Problems

A. Core Skill Development

Find $dy/dx$ using implicit differentiation. These problems involve single-rule applications and straightforward algebraic isolation.

1. $x^2 + y^2 = 49$
2. $3x + 5y = 15$
3. $x^3 + y^3 = 8$
4. $x^2 - y^2 = 1$ (unit hyperbola)
5. $\sqrt{x} + \sqrt{y} = 4$
6. $y^4 = x^3$
7. $x^2 y = 10$
8. $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$ (ellipse)
9. $x^{1/3} + y^{1/3} = 1$
10. $y^2 + 2y = x$

B. Structural Recognition

Each problem contains a structural feature — product of $x$ and $y$, a quotient, or a nested composition — that must be identified before differentiating. Write out your rule identification before computing.

1. $xy^2 + x^2 y = 6$
2. $x^2 e^y = y + 1$
3. $\ln(xy) = x - y$
4. $y\sin x = x^2 - y^2$
5. $\tan(y) = x^2 + 1$
6. $e^y + ye^x = 3$
7. $\cos(x - y) = y^2$
8. $\sqrt{xy} = x - y$
9. $x\ln y = y^2 - 1$
10. $\dfrac{y}{x+y} = x^2$

C. Multi-Step Problems

These problems involve multiple rules applied simultaneously, or require more careful algebraic manipulation to isolate $dy/dx$. Work systematically.

1. $\sin(xy) + \cos(x+y) = 1$
2. $x^2 y^3 - 3xy = 4$
3. $e^{x+y} = x^2 + y^2$
4. $\arctan(y/x) = \ln\sqrt{x^2+y^2}$
5. $x^3 + y^3 = 3xy + 1$ (find $dy/dx$ and points of horizontal tangency)
6. $(x^2 + y^2)^2 = 4xy$ (lemniscate-like curve; find slope at $(1,1)$)
7. $y = x^y$ (use logarithmic + implicit differentiation)
8. $\sin^2 x + \cos^2 y = 1$
9. $\arcsin(y) = x^2 \arccos(y)$
10. $\ln(x^2 + y) = xy^2$

D. Second Derivatives and Concavity

Find $d^2y/dx^2$ and, where specified, interpret the concavity geometrically.

1. $4x^2 + 9y^2 = 36$ (ellipse). Find $d^2y/dx^2$ and verify that it is negative on the upper half.
2. $xy = 4$ (hyperbola). Find $d^2y/dx^2$.
3. $x^3 - y^3 = 1$. Find $d^2y/dx^2$.
4. $\sin y = x$. Find $d^2y/dx^2$ in terms of $y$.
5. $y^2 = x^3$ (semi-cubical parabola). Find $d^2y/dx^2$ and determine where the curve is concave up.

E. Tangent Lines, Normal Lines, and Curve Analysis

Recall: a normal line at a point is perpendicular to the tangent line at that point.

1. Find the equation of the tangent line to $x^2 + xy + y^2 = 7$ at the point $(2, 1)$.
2. Find the equation of the normal line to $y^2 = x^3 - x$ at $(2, \sqrt{6})$.
3. At what points does $\dfrac{x^2}{4} + y^2 = 1$ have vertical tangent lines?
4. Find all points on $x^2 - 2xy + 4y^2 = 4$ where the tangent line is horizontal.
5. Show that the tangent line to $x^{2/3} + y^{2/3} = a^{2/3}$ (astroid) at point $(x_0, y_0)$ has $x$-intercept $a^{2/3}x_0^{1/3}$ and $y$-intercept $a^{2/3}y_0^{1/3}$.

F. Exam-Level Questions

These problems combine multiple skills or involve reasoning steps that are commonly tested under time pressure.

1. Let $F(x,y) = x^3 y - y^3 x = 0$ define $y$ implicitly as a function of $x$. (a) Find $dy/dx$. (b) Find $d^2y/dx^2$. (c) Determine whether the curve is concave up or down at the point $(1, 0)$ — or explain why the test fails.
2. The curve $C$ is defined by $e^y - y = x^2 - 1$. (a) Show that the point $(1, 0)$ lies on $C$. (b) Find the slope of $C$ at $(1, 0)$. (c) Find the second derivative at $(1, 0)$ and classify the concavity.
3. Suppose $x$ and $y$ satisfy $x^2 + 4xy - y^2 = 8$. Using implicit differentiation, find all values of $x$ at which the curve has a horizontal tangent, assuming real solutions exist.
4. (Free Response style) The equation $x \sin y + y \cos x = \pi/2$ implicitly defines $y$ as a function of $x$ near the point $\left(\pi/2, \pi/2\right)$. (a) Verify the point is on the curve. (b) Find the slope of the tangent line. (c) Write the equation of the tangent line.
5. (Multiple concept) A particle moves along the curve $x^2 y^2 = 16$. At the point $(2, 2)$, $x$ is increasing at $3$ units/sec. Find the rate of change of $y$ at that instant. (Differentiate implicitly with respect to time $t$.)

G. Challenge Set

For students who want to push beyond standard course expectations. These problems require deeper structural insight or more involved algebra.

1. For $x^4 + y^4 = 1$, find the coordinates of all points where the curvature is maximized. (Curvature $\kappa = \dfrac{|y''|}{(1+(y')^2)^{3/2}}$; first find $y'$ and $y''$ implicitly.)
2. The equation $y^5 + y = x$ cannot be solved for $y$ explicitly. Find a formula for $d^ny/dx^n$ in terms of $y$ and lower derivatives, for $n = 1, 2, 3$.
3. Show that if $F(x,y) = 0$ defines $y$ as a differentiable function of $x$, then $\dfrac{dy}{dx} = -\dfrac{F_x}{F_y}$, where $F_x$ and $F_y$ are partial derivatives. Apply this to $F = x^2 + 3xy - y^3 + 7$ and verify your result against the standard implicit method.

Error Diagnosis Problems

Each problem below shows a student's incorrect work. Identify the error, explain why it is wrong, and produce the correct solution.

1. Student's work: For $x^2 + y^2 = 1$, they wrote $2x + 2y = 0$, then $y = -x$.
   Identify the error and correct it.
2. Student's work: For $xy = 5$, they wrote $\dfrac{d}{dx}[xy] = 1 \cdot 1 = 1$, so $1 = 0$, a contradiction.
   Identify the error and correct it.
3. Student's work: For $\cos(y) = x^2$, they wrote $-\sin(y) = 2x$, giving $y = -\arcsin(2x)$.
   Is this correct? If not, identify the error.

Conceptual Reasoning Questions

These questions test understanding rather than computation. Full sentences are expected.

1. Explain in words (without formulas) why differentiating $y^3$ with respect to $x$ produces $3y^2 \dfrac{dy}{dx}$ rather than just $3y^2$. What principle is at work?
2. A student claims: "If a curve has a horizontal tangent at a point, then $dy/dx = 0$ at that point, so the numerator of my $dy/dx$ expression must be zero — but I don't need to check whether the denominator is also zero." Is this reasoning sound? Construct a counterexample or confirm it.
3. Why does the result of implicit differentiation typically contain both $x$ and $y$, rather than just $x$ alone? What does this mean geometrically when evaluating slope at a specific point?
4. Suppose you correctly find $\dfrac{dy}{dx} = \dfrac{2x - y}{x - 2y}$. At what points on the curve does the formula fail to be defined, and what geometric feature do those points represent?
5. The equation $x^2 + y^2 = -4$ has no real solutions. What happens if you try to apply implicit differentiation to it? What does the resulting expression for $dy/dx$ tell you, if anything?

Real-World Modeling Problems

Physics: Planetary Orbit

The orbit of a planet around the Sun is modeled by the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, where $a$ and $b$ are the semi-major and semi-minor axes in AU (astronomical units).

1. Show that the slope of the orbit at any point $(x,y)$ is $\dfrac{dy}{dx} = -\dfrac{b^2 x}{a^2 y}$.
2. For Earth's orbit, $a = 1.000$ AU and $b = 0.9998$ AU. At the point where $x = 0.5$ AU and $y > 0$, find $y$ (to 4 decimal places) and then compute the slope of Earth's orbital path. Interpret the sign of the slope in the context of Earth's motion.

Engineering: Implicit Constraint Surface

In a pressurized gas system, the relationship between pressure $P$ (in atm) and volume $V$ (in liters) for a Van der Waals gas satisfies:

where $a$, $b$, $n$, $R$, $T$ are constants.

1. Differentiate both sides with respect to $V$, treating $P$ as an implicitly defined function of $V$. Solve for $\dfrac{dP}{dV}$.
2. Interpret the sign of $\dfrac{dP}{dV}$ physically. Under what algebraic condition (on $a$, $b$, $P$, $V$) would $\dfrac{dP}{dV} = 0$? What physical state does this represent?

Economics: Production Isoquant

A firm's production of $Q$ units is modeled by the Cobb-Douglas isoquant $K^{0.4} L^{0.6} = Q_0$, where $K$ is capital and $L$ is labor, and $Q_0$ is a fixed output level.

1. Find $\dfrac{dK}{dL}$ (the marginal rate of technical substitution, MRTS) using implicit differentiation. Express in terms of $K$ and $L$.
2. Interpret the MRTS economically: what does $\dfrac{dK}{dL} = -2$ mean in this context, and why is it negative?
3. At the production point $K = 100$, $L = 200$, compute the MRTS numerically.

Biology: Population Constraint

Two competing species have population levels $x$ and $y$ (in thousands) that satisfy the constraint $x^2 + xy + y^2 = 3Q$ for some ecological constant $Q > 0$.

1. Find $\dfrac{dy}{dx}$ and interpret it as the rate at which species $y$'s population changes with respect to species $x$'s population along the ecological boundary.
2. At what population ratios $x:y$ is $\dfrac{dy}{dx} = -1$? What does this mean for the two species?

Optics: Refraction Curve

The boundary of an optical lens is described by the curve $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$. A light ray strikes the lens at the point $(3, 3.2)$.

1. Verify that $(3, 3.2)$ is on the lens curve (approximately). Find the slope of the tangent to the lens at that point. This tangent gives the local orientation of the lens surface.
2. Find the slope of the normal to the lens at that point. The normal determines the angle of incidence for the incoming light ray.

Exam Traps and Strategic Advice

Mixed Rule Detection

For each problem below, before computing: (i) name every differentiation rule you will need, and (ii) identify which terms require each rule. Then solve.

1. $x^2 \sin y + y^2 \cos x = 1$
2. $e^{xy} \ln(x + y) = 3$
3. $\sqrt{x^2 + y^2} = \arctan(y/x)$
4. $\left(\dfrac{x}{y}\right)^2 + \ln y = x$
5. $y^x = x^y$ (take logarithms of both sides first; treat both $y$ and $x$ as variable)
6. $\cos(e^y) = x^3 + y^2$
7. $\arcsin(xy) = \dfrac{\pi}{4}$
8. $x^2\arctan y - y^2\arctan x = 0$

Frequently Asked Questions