# Related Rates and Implicit Differentiation

Mathematics I · Differential Calculus · https://tryals.app/learn/mathematics-i/related-rates-and-implicit-differentiation

## Implicit Differentiation

Not every curve is written as $y = f(x)$. An **implicit** relation like $x^2 + y^2 = 25$ defines $y$ through $x$ without solving for it explicitly.

**Implicit differentiation** finds $\frac{dy}{dx}$ by differentiating both sides with respect to $x$. Treat $y$ as a function of $x$ and apply the **chain rule** to every $y$-term.

| Relation | Action | Result |
| --- | --- | --- |
| $y^2$ | $\frac{d}{dx}$ | $2y\frac{dy}{dx}$ |
| $x^2 + y^2 = 25$ | $\frac{d}{dx}$ | $2x + 2y\frac{dy}{dx} = 0$ |

Solving yields $\frac{dy}{dx} = -\frac{x}{y}$. This handles tangled curves that resist explicit isolation.

## Related Rates

When quantities link up and change over **time $t$**, their rates link up too. Differentiate with respect to $t$, bringing in a $\frac{d}{dt}$ factor for every changing variable.

Method: Write equation $\rightarrow$ Differentiate wrt $t$ $\rightarrow$ Substitute and solve.

For a sphere ($V = \frac{4}{3}\pi r^3$), differentiating gives $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$.

> **Common pitfall:** Forgetting the **chain rule** factor (like $\frac{dy}{dx}$ or $\frac{dr}{dt}$) on dependent variables. Treating $y^2$ as just $2y$ is the classic fatal error.

## Practice questions

4 of this lesson's 9 practice questions, with answers. The full set is in the app.

### 1. Differentiating y² with respect to x gives 2y·(dy/dx), not 2y. Why does the extra factor appear?

A. Because y is a function of x, so y² is a composition.
B. Because y² must be rewritten in terms of x before it can be differentiated, and dy/dx records that rewriting.
C. Because the power rule requires an extra factor whenever the variable is not named x.
D. Because dy/dx is a notational convention that is always attached when y appears.

**Answer:** A. Because y is a function of x, so y² is a composition.

**Why:** Implicit differentiation exists for the case where y is defined by an equation you cannot solve, so y is a function of x and y² is really (y(x))², a composition, and dropping the factor treats y as constant, contradicting the setup. Check it on the circle: both routes give −2x, and only because the chain-rule factor was kept.

Page: https://tryals.app/practice/mathematics-i/related-rates-and-implicit-differentiation/differentiating-y-with-respect-to-x-gives-2y-dy-dx-not-2y-why

### 2. Order the steps of solving a related-rates problem.

**Answer:**

1. Write an equation relating the quantities involved
2. Differentiate both sides with respect to time
3. Apply the chain rule so that every changing variable contributes its own rate
4. Substitute the known values and known rates
5. Solve for the unknown rate

**Why:** Substituting before differentiating is the fatal shortcut: putting r = 2 into the volume formula first makes V a constant, so dV/dt would come out as 0. Numbers go in only AFTER the differentiation is complete.

Page: https://tryals.app/practice/mathematics-i/related-rates-and-implicit-differentiation/order-the-steps-of-solving-a-related-rates-problem

### 3. A ladder slides down a wall. Its top falls faster and faster while the base moves out at a steady speed. Why does a constant horizontal rate produce an accelerating vertical one?

A. Because the base actually speeds up too, which the description omits
B. Because x² + y² = L² ties them, and dy/dt grows as y shrinks
C. Because the ladder shortens as it slides, changing the relationship
D. Because gravity accelerates the top of the ladder independently of the base

**Answer:** B. Because x² + y² = L² ties them, and dy/dt grows as y shrinks

**Why:** The constraint does the work: differentiating x² + y² = L² ties the two rates, and as y → 0 the vertical rate must blow up. The model predicts infinite speed, which the physical ladder answers by leaving the wall, a good reminder that a constraint holds only while its assumptions do.

Page: https://tryals.app/practice/mathematics-i/related-rates-and-implicit-differentiation/a-ladder-slides-down-a-wall-its-top-falls-faster-and-faster-while

### 4. y is a function of x. Select every derivative with respect to x that is CORRECT.

A. The chain rule is not needed when differentiating implicitly.
B. Differentiating y squared with respect to x brings a dy/dx factor with it.
C. Implicit differentiation avoids solving the equation for y first.
D. In related rates, every changing quantity carries a d/dt factor.
E. The derivative of x squared times y is 2xy plus x squared times dy/dx.

**Answer:** B. Differentiating y squared with respect to x brings a dy/dx factor with it.; D. In related rates, every changing quantity carries a d/dt factor.; E. The derivative of x squared times y is 2xy plus x squared times dy/dx.

**Why:** Options 3 and 5 both drop a dy/dx: the correct results are cos y times dy/dx, and y squared plus 2xy dy/dx. Option 2 shows the product rule applied correctly, with the chain rule supplying the dy/dx only on the y-factor.

Page: https://tryals.app/practice/mathematics-i/related-rates-and-implicit-differentiation/y-is-a-function-of-x-select-every-derivative-with-respect-to-x-that