9.3 Derivatives

Definition of derivatives

A derivative tells you how fast a function's output is changing at any given input. It's the central idea in differential calculus, and it shows up everywhere: velocity in physics, marginal cost in economics, slope on a graph. If you understand derivatives well, you have a tool that works across many fields.

Instantaneous rate of change

The instantaneous rate of change is the rate at which a function is changing at one specific point, not over an interval. Think of it this way: your car's speedometer shows your speed right now, not your average speed over the whole trip. That speedometer reading is an instantaneous rate of change.

Mathematically, you get it by taking the average rate of change over a shrinking interval and seeing what it approaches as that interval gets infinitely small. This is what separates it from the average rate of change, which uses a finite interval between two points.

Slope of tangent line

The derivative at a point is the same thing as the slope of the tangent line to the curve at that point. A tangent line just touches the curve at one spot and shows the direction the curve is heading right there.

• A positive slope means the function is increasing at that point
• A negative slope means it's decreasing
• A slope of zero means the function is momentarily flat, which often signals a local maximum or minimum

Limit definition

The formal definition of the derivative is:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Here's what's happening: $\frac{f(x+h) - f(x)}{h}$ is the slope of a secant line connecting two points on the curve that are $h$ apart. As $h$ shrinks toward zero, that secant line rotates into the tangent line, and its slope becomes the derivative. Every derivative rule you'll learn is ultimately derived from this definition.

Derivative rules

These rules let you find derivatives quickly without going back to the limit definition each time. Once you know them, you can differentiate most functions in seconds.

Power rule

For any function $f(x) = x^n$ where $n$ is a real number:

$\frac{d}{dx}(x^n) = nx^{n-1}$

Multiply by the exponent, then reduce the exponent by one. This works for negative and fractional exponents too. For example, $\frac{d}{dx}(x^3) = 3x^2$, and $\frac{d}{dx}(x^{-2}) = -2x^{-3}$.

Product rule

When you need the derivative of two functions multiplied together:

$(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)$

In words: differentiate the first and multiply by the second, then add the first times the derivative of the second. This saves you from having to expand complicated products before differentiating.

Quotient rule

For the derivative of one function divided by another:

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

A common mnemonic: "low d-high minus high d-low, over the square of what's below." Be careful with signs here; mixing up the subtraction order is one of the most common mistakes.

Chain rule

The chain rule handles composite functions, where one function is nested inside another:

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

You differentiate the outer function (leaving the inner function untouched), then multiply by the derivative of the inner function. For example, to differentiate $(3x+1)^5$, treat the outer function as $u^5$ and the inner as $u = 3x+1$. You get $5(3x+1)^4 \cdot 3 = 15(3x+1)^4$.

Common derivatives

Memorizing these saves significant time. They're the building blocks you'll combine with the rules above.

Trigonometric functions

• $\frac{d}{dx}(\sin x) = \cos x$
• $\frac{d}{dx}(\cos x) = -\sin x$
• $\frac{d}{dx}(\tan x) = \sec^2 x$

Notice the negative sign on cosine's derivative. That trips people up frequently.

Exponential functions

• $\frac{d}{dx}(e^x) = e^x$ (the natural exponential is its own derivative)
• $\frac{d}{dx}(a^x) = a^x \ln a$ for any positive constant $a$

These show up constantly in growth and decay models, like population dynamics or radioactive decay.

Logarithmic functions

• $\frac{d}{dx}(\ln x) = \frac{1}{x}$
• $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$

The logarithmic derivatives are the inverses of the exponential ones, which makes sense given that logarithms and exponentials are inverse functions.

Applications of derivatives

This is where derivatives become more than abstract math. These applications show up in physics, engineering, business, and beyond.

Optimization problems

Optimization means finding the maximum or minimum value of a function. The process:

1. Write a function that models what you want to maximize or minimize
2. Find critical points by setting $f'(x) = 0$ (or finding where $f'(x)$ is undefined)
3. Use the second derivative test: if $f''(x) > 0$ at a critical point, it's a local minimum; if $f''(x) < 0$, it's a local maximum
4. Check endpoints of the domain if the function is defined on a closed interval

Real-world examples include maximizing profit, minimizing material usage in manufacturing, or finding the dimensions that give the largest area for a given perimeter.

Related rates

Related rates problems ask: if one quantity is changing at a known rate, how fast is a related quantity changing? The general approach:

1. Identify all changing quantities and write an equation relating them
2. Differentiate both sides with respect to time $t$ (using implicit differentiation and the chain rule)
3. Plug in known values and solve for the unknown rate

Classic examples: a ladder sliding down a wall, water filling a cone-shaped tank, or a balloon expanding as air is pumped in.

Motion analysis

Derivatives connect position, velocity, and acceleration:

• Position: $s(t)$
• Velocity: $v(t) = s'(t)$ (first derivative of position)
• Acceleration: $a(t) = v'(t) = s''(t)$ (second derivative of position)

If velocity is positive, the object moves in the positive direction. If acceleration and velocity have the same sign, the object is speeding up; opposite signs mean it's slowing down.

Higher-order derivatives

Taking the derivative of a derivative gives you higher-order derivatives, which reveal additional information about a function's behavior.

Second derivative

The second derivative, written $f''(x)$ or $\frac{d^2y}{dx^2}$, tells you the rate of change of the first derivative. In physical terms, if $s(t)$ is position, the first derivative is velocity and the second derivative is acceleration.

The second derivative also describes curvature: how quickly the slope itself is changing.

Concavity and inflection points

The sign of the second derivative tells you about the shape of the curve:

• $f''(x) > 0$: the curve is concave up (shaped like a cup, bending upward)
• $f''(x) < 0$: the curve is concave down (shaped like a cap, bending downward)

An inflection point is where the concavity switches. To find inflection points, solve $f''(x) = 0$, then verify that the sign of $f''(x)$ actually changes on either side of that point. (Just because $f''(x) = 0$ doesn't guarantee an inflection point; you need a sign change.)

Implicit differentiation

Definition and process

Sometimes you can't easily solve an equation for $y$ in terms of $x$. For example, the equation of a circle $x^2 + y^2 = 25$ defines $y$ implicitly. Implicit differentiation lets you find $\frac{dy}{dx}$ without isolating $y$ first.

The process:

1. Differentiate both sides of the equation with respect to $x$
2. Every time you differentiate a term containing $y$, apply the chain rule and multiply by $\frac{dy}{dx}$ (since $y$ is a function of $x$)
3. Collect all terms with $\frac{dy}{dx}$ on one side
4. Solve for $\frac{dy}{dx}$

The result will typically be an expression involving both $x$ and $y$.

Applications in physics

Implicit differentiation is used in thermodynamics (where state variables like pressure, volume, and temperature are related by equations that aren't easily solved for one variable), constrained motion problems in mechanics, and electromagnetic theory. Any time you have interdependent variables connected by a relationship, implicit differentiation can help you find how one changes relative to another.

Derivative notation

Two main notation systems exist, and you'll see both throughout your studies.

Leibniz notation

Writes the derivative as $\frac{dy}{dx}$, which emphasizes the idea of a ratio of infinitesimal changes in $y$ and $x$. This notation is especially helpful when applying the chain rule, since you can "cancel" differentials: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. It's the standard in physics and engineering.

Lagrange notation

Uses prime marks: $f'(x)$ for the first derivative, $f''(x)$ for the second, and so on. It's more compact and common in pure mathematics. For higher-order derivatives beyond the third, you'll often see $f^{(n)}(x)$ instead of stacking prime marks.

Continuity and differentiability

Relationship between concepts

A key fact: all differentiable functions are continuous, but not all continuous functions are differentiable. Differentiability is a stronger condition than continuity.

• Continuity means no breaks, jumps, or holes in the graph
• Differentiability means the function has a well-defined tangent line (a unique, finite slope) at every point

A function can be continuous at a point but fail to be differentiable there if it has a sharp corner, a cusp, or a vertical tangent line.

Discontinuities vs non-differentiability

Discontinuities are points where the function itself breaks:

• Removable: a hole in the graph (the limit exists but doesn't equal the function value)
• Jump: the function jumps from one value to another
• Infinite: the function blows up toward infinity (like $\frac{1}{x}$ at $x = 0$)

Non-differentiable points are where the function is continuous but has no single tangent line. The classic example is $f(x) = |x|$ at $x = 0$: the graph is continuous but has a sharp corner, so no unique tangent line exists there.

Derivative approximations

Linear approximation

Also called linearization, this uses the tangent line at a known point to estimate nearby function values:

$f(x) \approx f(a) + f'(a)(x - a)$

This works well when $x$ is close to $a$. For example, you could approximate $\sqrt{4.1}$ by using the tangent line to $f(x) = \sqrt{x}$ at $a = 4$. Linear approximation is the foundation for Taylor series, which extend this idea to higher-order polynomials for better accuracy.

Newton's method

An iterative technique for finding roots (where $f(x) = 0$):

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Starting from an initial guess $x_0$, each step uses the tangent line at the current point to get a better approximation of the root. For well-behaved functions with a reasonable starting guess, convergence is very fast. The method can fail if $f'(x_n) = 0$ at some step or if the initial guess is too far from the root.

Derivatives in economics

Marginal cost

Marginal cost is the derivative of the total cost function $C(q)$ with respect to quantity $q$. It tells you approximately how much it costs to produce one more unit. If $C(q) = 500 + 3q + 0.01q^2$, then $C'(q) = 3 + 0.02q$, so at $q = 100$ units, the marginal cost is $3 + 2 = 5$ dollars per unit. Businesses use this to decide optimal production levels: produce more as long as marginal revenue exceeds marginal cost.

Elasticity of demand

Price elasticity of demand measures how sensitive consumers are to price changes:

$E = \frac{dQ}{dP} \cdot \frac{P}{Q}$

• $|E| > 1$: elastic demand (consumers are very responsive to price changes; a small price increase causes a large drop in quantity demanded)
• $|E| < 1$: inelastic demand (consumers are less responsive; think necessities like gasoline)
• $|E| = 1$: unit elastic

This concept is central to pricing strategy and tax policy analysis.