Lecture 21: Multiplication and Division Rules

Watch 3Blue1Brown videos on Derivative formulas through geometry and visualizing the chain rule and product rule for a further conceptual understanding of the rules.

Constant Multiple Rule

If $f(x)$ is a differentiable function and $c$ is a constant, then:

$$\frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}[f(x)]$$

Example:

$\frac{d}{dx}[7x^2] = 7 \cdot \frac{d}{dx}[x^2] = 7 \cdot 2x = 14x$

This rule lets us factor out constants when taking derivatives.

Multiplication Rule (Product Rule)

If $f(x)$ and $g(x)$ are both differentiable, then:

$$\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)$$

“First times derivative of second, plus second times derivative of first.”

Example 1:

$f(x) = x^2$, $g(x) = x+1$
Then:

$$\frac{d}{dx}[(x^2)(x+1)] = 2x(x+1) + x^2(1) = 2x^2 + 2x + x^2 = 3x^2 + 2x$$

Example 2:

$\frac{d}{dx}[(x^2+1)(x^3-2x)]$
Apply the product rule:

• First: $f(x) = x^2+1$, so $f'(x) = 2x$
• Second: $g(x) = x^3 - 2x$, so $g'(x) = 3x^2 - 2$

So:

$$\frac{d}{dx}[(x^2+1)(x^3-2x)] = 2x(x^3 - 2x) + (x^2+1)(3x^2 - 2) = 2x^4 - 4x^2 + 3x^2 - 2 =$$

$$= 2x^4 - x^2 - 2$$

Division Rule (Quotient Rule)

If $f(x)$ and $g(x)$ are differentiable, and $g(x) \ne 0$, then:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}$$

“Low d-high minus high d-low over square of what’s below.”

Example 1:

$\frac{d}{dx}\left[\frac{x^2}{x+1}\right]$
Let $f(x) = x^2$, $g(x) = x+1$

Then:

$$\frac{2x(x+1) - x^2(1)}{(x+1)^2} = \frac{2x(x+1) - x^2}{(x+1)^2}$$

Example 2:

$\frac{d}{dx} \left[\frac{1}{x^2}\right]$
Use quotient rule:

$$\frac{0\cdot x^2 - 1\cdot 2x}{x^4} = \frac{-2x}{x^4} = -\frac{2}{x^3}$$

which will be the same answer if you just apply the power rule to $x^{-2}$:

$$\frac{d}{dx}[x^{-2}] = -2x^{-2 - 1} = -2x^{-3} = -\frac{2}{x^3}$$

With this rule, we can differentiate any rational function. For example,

$$\frac{d}{dx}\left[\frac{x^2 + 1}{x^2 - 1}\right] = \frac{(2x)(x^2 - 1) - (x^2 + 1)(2x)}{(x^2 - 1)^2} = \frac{2x^3 - 2x - 2x^3 - 2x}{(x^2 - 1)^2} = \frac{-4x}{(x^2 - 1)^2}$$

Summary Schedule: Product & Quotient Rules

Exercises

1. Constant multiple rule

Differentiate the following using constant multiple rule:

• a) $\frac{d}{dx}[5x^3]$
• b) $\frac{d}{dx}[-7\sqrt{x}]$
• c) $\frac{d}{dx}[3e^x]$

2. Product rule

Differentiate the following using the product rule:

• a) $\frac{d}{dx}[x \cdot \ln(x)]$
• b) $\frac{d}{dx}[e^x \cdot x^2]$
• c) $\frac{d}{dx}[(x^2 + 4x)(x^3 - 1)]$

3. Quotient rule

Differentiate the following using the quotient rule:

• a) $\frac{d}{dx}\left[\frac{1}{x}\right]$
• b) $\frac{d}{dx}\left[\frac{x^2 + 1}{x}\right]$
• c) $\frac{d}{dx}\left[\frac{\ln(x)}{x^2}\right]$

4. Additional Practice

Differentiate:

• a) $\frac{d}{dx}[x^2 \cdot \ln(3x)]$
• b) $\frac{d}{dx}\left[\frac{x^3 + 2x}{x^2 + 1}\right]$
• c) $\frac{d}{dx}\left[\frac{e^x}{x^3 + 2}\right]$
• d) $\frac{d}{dx}[(2x^2 + 3)(x^3 - 5x)]$
• e) $\frac{d}{dx}\left[\frac{x^2(x-1)}{x+2}\right]$
  Hint: Expand numerator before applying quotient rule if easier.

5. Evaluate

Let $f(x) = \frac{g(x)}{h(x)}$. Suppose $g(2) = 5$, $g'(2) = 3$, $h(2) = 1$, and $h'(2) = -4$.

• a) Find $f'(2)$.

6. Additional practice

Follow and solve the exercises in this link.

Extra Advanced Exercises

1. Proof of Product and Quotient Rules

Using the limit definition of derivatives, show that:
(a) $\frac{d}{dx}(f(x)g(x))=f'(x)g(x)+f(x)g'(x)$
(b) $\frac{d}{dx}(\frac{f(x)}{g(x)})=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}$

2. Water Flow in a Conical Container

Water is being poured into a conical tank such that the volume added per second is related to the height of the water $x$ (in meters) and the radius of the water surface $r$ by:

$$f(x) = \frac{x^2 \ln(x)}{x + 1}$$

• Find how fast the volume is changing with respect to the height.

3. Nested Quotient of Products

Let

$$f(x) = \frac{(x^2 + 1)(x - 4)}{(x + 2)(x^2 - 1)}$$

• (a) Differentiate $f(x)$
• (b) Simplify and identify where $f'(x) = 0$

4. Pressure–Volume Constraint in a Gas

The gas pressure and volume are related by:

$$\ln(xy) = \frac{x}{y}$$

• Use implicit differentiation (and logarithmic identities) to find how volume $y$ changes as pressure $x$ increases.

5. Two Interrelated Functions

Let:

$$f(x) = \frac{g(x)}{h(x)}, \quad \text{and } g(x)h(x) = x^2 + 1$$

• Use implicit differentiation on the product identity to find $f'(x)$.