12.4 Derivatives

Derivatives and Their Applications

The derivative of a function tells you how fast that function's output is changing at any given input. If you know the derivative, you can find slopes of curves, analyze motion, and locate maximum and minimum values of functions.

This section covers the core derivative rules, how to write tangent line equations, motion analysis, and how derivatives connect to finding extrema.

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Derivatives of various functions

A derivative gives you a new function that outputs the slope of the original function at every point. The notation $f'(x)$ means "the derivative of $f(x)$." Here are the rules you need to know.

Power Rule: For any function of the form $f(x) = x^n$, bring the exponent down as a coefficient and reduce the exponent by 1:

$f'(x) = nx^{n-1}$

• $f(x) = x^4$ has derivative $f'(x) = 4x^3$
• $f(x) = 3x^5$ has derivative $f'(x) = 15x^4$

Constant Multiple Rule: If a function is multiplied by a constant, the constant stays and you differentiate the rest.

• $f(x) = 4x^3$ has derivative $f'(x) = 4 \cdot 3x^2 = 12x^2$
• $f(x) = -2x^2$ has derivative $f'(x) = -2 \cdot 2x = -4x$

Sum and Difference Rules: Differentiate each term separately, then add or subtract the results. The derivative of a constant term is always 0.

• $f(x) = x^3 + 2x^2$ has derivative $f'(x) = 3x^2 + 4x$
• $f(x) = x^4 - x^2$ has derivative $f'(x) = 4x^3 - 2x$
• $f(x) = 2x^3 - 3x + 1$ has derivative $f'(x) = 6x^2 - 3$

Exponential Function Rule: The function $f(x) = e^x$ is its own derivative: $f'(x) = e^x$. When the exponent has a coefficient (like $e^{kx}$), that coefficient multiplies the result. This uses the chain rule, which you'll study more in calculus, but for now just know:

• $f(x) = e^x$ has derivative $f'(x) = e^x$
• $f(x) = 3e^{2x}$ has derivative $f'(x) = 3 \cdot 2e^{2x} = 6e^{2x}$

Derivative as instantaneous change

The derivative at a point gives you the instantaneous rate of change of the function at that exact input. Think of it as the slope of the curve at a single point, rather than an average rate over an interval.

This idea shows up across many fields:

• Velocity is the derivative of position. It tells you how fast an object's position is changing at a specific moment.
• Acceleration is the derivative of velocity. It tells you how fast the velocity itself is changing.
• Marginal cost is the derivative of total cost. It approximates the additional cost of producing one more unit.
• Marginal revenue is the derivative of total revenue. It approximates the additional revenue from selling one more unit.
• Population growth rate is the derivative of population size with respect to time, measuring how quickly a population is changing at a given moment.

Derivatives are also central to optimization problems, where you find the input values that produce the largest or smallest output of a function.

Tangent line equations

A tangent line touches a curve at exactly one point and has the same slope as the curve at that point. The equation uses point-slope form:

$y - f(a) = f'(a)(x - a)$

where $a$ is the x-value of the point of tangency.

Steps to find a tangent line equation:

1. Differentiate $f(x)$ to get $f'(x)$
2. Plug in $x = a$ to $f'(x)$ to find the slope $f'(a)$
3. Plug in $x = a$ to $f(x)$ to find the y-coordinate $f(a)$
4. Substitute the point $(a, f(a))$ and slope $f'(a)$ into point-slope form

Example: Find the tangent line to $f(x) = x^2$ at $x = 3$.

1. $f'(x) = 2x$
2. $f'(3) = 2(3) = 6$, so the slope is 6
3. $f(3) = 3^2 = 9$, so the point is $(3, 9)$
4. Tangent line: $y - 9 = 6(x - 3)$, which simplifies to $y = 6x - 9$

Derivatives in motion analysis

Position, velocity, and acceleration form a derivative chain:

• Position $s(t)$ describes where an object is at time $t$
• Velocity $v(t) = s'(t)$ is the first derivative of position
• Acceleration $a(t) = v'(t) = s''(t)$ is the second derivative of position

The signs of these quantities tell you about the motion:

• Positive velocity means the object moves in the positive direction (right or up); negative velocity means the opposite direction
• When velocity and acceleration have the same sign, the object is speeding up. When they have opposite signs, the object is slowing down.

Example: A ball is thrown upward with position $s(t) = -4.9t^2 + 20t$ (meters). Find velocity and acceleration at $t = 2$ seconds.

1. $v(t) = s'(t) = -9.8t + 20$. At $t = 2$: $v(2) = -9.8(2) + 20 = 0.4$ m/s (positive, so still rising)
2. $a(t) = v'(t) = -9.8$ m/s² (constant, pulling downward)
3. At $t = 2$, velocity is positive but acceleration is negative. They have opposite signs, so the ball is slowing down.

Continuity, Differentiability, and Local Extrema

Continuity means the function has no breaks, jumps, or holes. A function must be continuous at a point before it can be differentiable there. However, continuity alone doesn't guarantee differentiability. Sharp corners and cusps are continuous but not differentiable because there's no single well-defined tangent line.

Differentiability at a point means the function has a well-defined derivative (and therefore a smooth tangent line) at that point.

Local extrema are the local maximum and minimum values of a function. They occur at critical points, which are x-values where:

• $f'(x) = 0$ (the tangent line is horizontal), or
• $f'(x)$ is undefined

To determine whether a critical point is a local max or min, check how the sign of $f'(x)$ changes around that point:

• If $f'(x)$ changes from positive to negative, you have a local maximum (the function was increasing, then starts decreasing).
• If $f'(x)$ changes from negative to positive, you have a local minimum (the function was decreasing, then starts increasing).
• If $f'(x)$ doesn't change sign, the critical point is neither a max nor a min.

These ideas are the foundation of optimization: find the critical points, then determine which gives the largest or smallest value.