Calculus I · Chapter 4
Chapter 4: Calculating the Derivative
Power rule · Marginal analysis · Product & quotient rules · The chain rule · Exponential & logarithmic derivatives
The limit definition works, but it's slow. This chapter replaces it with a small set of differentiation rules that turn finding derivatives into pure algebra. Master these and you can differentiate almost any function in the course on sight, including the exponential and logarithmic functions, whose derivatives are remarkably clean.
Section 4.1
Techniques for Finding Derivatives
The Power Rule and Its Companions
The Core Rules
| Rule | Statement |
|---|---|
| Constant | $\dfrac{d}{dx}[c] = 0$ |
| Power | $\dfrac{d}{dx}[x^n] = n x^{n-1}$ (any real $n$) |
| Constant multiple | $\dfrac{d}{dx}[c\,f(x)] = c\,f'(x)$ |
| Sum / difference | $\dfrac{d}{dx}[f \pm g] = f' \pm g'$ |
The power rule as a pattern. Step the exponent $n$ and watch $x^n \to n\,x^{n-1}$: the old exponent drops down in front, and the new exponent is one lower. The same mechanical move every time.
⚠ Rewrite Before You Differentiate
The power rule needs a clean power of $x$. Rewrite roots and fractions first:
$$\sqrt{x} = x^{1/2}, \qquad \frac{1}{x^3} = x^{-3}, \qquad \frac{1}{\sqrt[3]{x}} = x^{-1/3}.$$
Then apply the rule. Negative and fractional exponents are completely allowed.
📘 Example 4.1: Power Rule with Mixed Terms
Differentiate $f(x) = 4x^3 - \dfrac{2}{x} + 5\sqrt{x} - 7$.
Rewrite as powers
$f(x) = 4x^3 - 2x^{-1} + 5x^{1/2} - 7$.
Differentiate term by term
$$f'(x) = 12x^2 + 2x^{-2} + \tfrac{5}{2}x^{-1/2} = 12x^2 + \frac{2}{x^2} + \frac{5}{2\sqrt{x}}.$$
$f'(x) = 12x^2 + \dfrac{2}{x^2} + \dfrac{5}{2\sqrt{x}}$
Marginal Analysis
Economic Meaning of the Derivative
In business, the derivative of a total function is the marginal function, the approximate change from one more unit:
- Marginal cost $C'(x)$ ≈ cost of producing the next item.
- Marginal revenue $R'(x)$ ≈ revenue from selling the next item.
- Marginal profit $P'(x) = R'(x) - C'(x)$.
📘 Example 4.2: Marginal Cost
A cost function is $C(x) = 0.01x^2 + 5x + 100$ dollars. Find the marginal cost at $x = 50$ and interpret it.
Differentiate, then evaluate
$C'(x) = 0.02x + 5$, so $C'(50) = 0.02(50) + 5 = 6$.
The $51$st item costs about \$6 to produce.
$C'(50) = \$6$ per additional item
Section 4.2
Derivatives of Products & Quotients
Product & Quotient Rules
$$\textbf{Product:}\quad (fg)' = f'g + fg'$$
$$\textbf{Quotient:}\quad \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$$
The product rule is symmetric; the quotient rule is not, order matters, and the denominator gets squared.
⚠ The Derivative of a Product Is NOT the Product of Derivatives
$(fg)' \neq f'g'$. You must use the full product rule. A quick mnemonic for the quotient rule: "low d-high minus high d-low, over low squared."
📘 Example 4.3: Product Rule
Differentiate $y = (3x^2 + 1)(x^3 - 4x)$.
Let $f = 3x^2+1$, $g = x^3-4x$
$f' = 6x$, $g' = 3x^2 - 4$. Then
$$y' = f'g + fg' = 6x(x^3 - 4x) + (3x^2+1)(3x^2-4).$$
Expanding: $y' = 6x^4 - 24x^2 + 9x^4 - 12x^2 + 3x^2 - 4 = 15x^4 - 33x^2 - 4$.
$y' = 15x^4 - 33x^2 - 4$
📘 Example 4.4: Quotient Rule
Differentiate $y = \dfrac{2x - 1}{x^2 + 3}$.
$f = 2x-1$, $g = x^2+3$
$f' = 2$, $g' = 2x$. Then
$$y' = \frac{f'g - fg'}{g^2} = \frac{2(x^2+3) - (2x-1)(2x)}{(x^2+3)^2} = \frac{2x^2 + 6 - 4x^2 + 2x}{(x^2+3)^2}.$$
$$y' = \frac{-2x^2 + 2x + 6}{(x^2+3)^2}.$$
$y' = \dfrac{-2x^2 + 2x + 6}{(x^2+3)^2}$
Sponsored
Section 4.3
The Chain Rule
The chain rule differentiates composite functions, a function inside another function, like $(3x + 1)^5$ or $\sqrt{x^2 + 4}$. It is the most-used rule in all of calculus.
Theorem: The Chain Rule
If $y = f(g(x))$, then
$$\frac{dy}{dx} = f'(g(x)) \cdot g'(x).$$
In Leibniz notation, with $y = f(u)$ and $u = g(x)$:
$$\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}.$$
Differentiate the outside (keeping the inside intact), then multiply by the derivative of the inside.
The chain rule as a machine. Input $x$ flows through the inner function $u = g(x)$, then the outer $y = f(u)$. The overall rate is the product of the two rates along the chain, outer rate times inner rate. Click to send a pulse through.
The "Generalized Power Rule": Most Common Case
$$\frac{d}{dx}\big[g(x)\big]^n = n\big[g(x)\big]^{n-1}\cdot g'(x).$$
Power rule on the outside, then times the derivative of the inside.
📘 Example 4.5: Generalized Power Rule
Differentiate $y = (x^2 + 3x)^4$.
Outer is $(\;)^4$; inner is $x^2 + 3x$
$$y' = 4(x^2+3x)^3 \cdot (2x + 3).$$
$y' = 4(2x+3)(x^2+3x)^3$
📘 Example 4.6: Chain Rule with a Root
Differentiate $y = \sqrt{x^2 + 1}$.
Rewrite: $y = (x^2+1)^{1/2}$
$$y' = \tfrac{1}{2}(x^2+1)^{-1/2}\cdot 2x = \frac{x}{\sqrt{x^2+1}}.$$
$y' = \dfrac{x}{\sqrt{x^2+1}}$
Section 4.4
Derivatives of Exponential Functions
Exponential Derivatives
$$\frac{d}{dx}\,e^x = e^x \qquad\text{(it is its own derivative!)}$$
With the chain rule, for an inner function $g(x)$:
$$\frac{d}{dx}\,e^{g(x)} = e^{g(x)}\cdot g'(x).$$
For a general base: $\dfrac{d}{dx}\,a^x = a^x \ln a$.
Why $e$ Is Special
$e^x$ is the unique exponential function equal to its own slope at every point. That self-replicating property is exactly why $e$ governs natural growth and decay, the rate of change is always proportional to the current amount.
📘 Example 4.7: Exponential Chain Rule
Differentiate $y = e^{3x^2 - 5x}$.
Inner $g = 3x^2 - 5x$, so $g' = 6x - 5$
$$y' = e^{3x^2 - 5x}\cdot(6x - 5).$$
$y' = (6x - 5)e^{3x^2 - 5x}$
Section 4.5
Derivatives of Logarithmic Functions
Logarithmic Derivatives
$$\frac{d}{dx}\,\ln x = \frac{1}{x} \qquad (x > 0)$$
With the chain rule:
$$\frac{d}{dx}\,\ln g(x) = \frac{g'(x)}{g(x)}.$$
For a general base: $\dfrac{d}{dx}\,\log_a x = \dfrac{1}{x\ln a}$.
📘 Example 4.8: Logarithmic Chain Rule
Differentiate $y = \ln(x^2 + 4x + 1)$.
Inner over itself
$$y' = \frac{2x + 4}{x^2 + 4x + 1}.$$
$y' = \dfrac{2x+4}{x^2+4x+1}$
📘 Example 4.9: Combining Several Rules
Differentiate $y = x^2 e^{x}$.
Product rule, with $f = x^2$, $g = e^x$
$f' = 2x$, $g' = e^x$. Then
$$y' = 2x\,e^x + x^2 e^x = x e^x(2 + x).$$
$y' = xe^x(x + 2)$
Master Reference: Differentiation Rules
| Function | Derivative |
|---|---|
| $c$ | $0$ |
| $x^n$ | $nx^{n-1}$ |
| $e^x$ | $e^x$ |
| $a^x$ | $a^x\ln a$ |
| $\ln x$ | $1/x$ |
| $f \cdot g$ | $f'g + fg'$ |
| $f / g$ | $(f'g - fg')/g^2$ |
| $f(g(x))$ | $f'(g(x))\cdot g'(x)$ |
Sponsored
Looking Ahead
You can now differentiate fluently. Chapter 5 uses the derivative to read a function, where it rises and falls, where it peaks, and how it curves, and to draw complete, accurate graphs from sign information alone.