Derivatives of Polynomials & Exponentials: Rules, Examples, and Applications

Section 3.1: Derivatives of Polynomials & Exponentials

I. The Constant & Power Rules for Derivatives

This section introduces the foundational rules for computing derivatives of polynomial and exponential functions, including the constant rule, identity rule, and power rule.

• Derivative of a Constant: The derivative of any constant function is zero. $\frac{d}{dx}(c) = 0$ Example: If $f(x) = 3$, then $f'(x) = 0$ (the slope of a constant function is always zero).

• Derivative of the Identity Function: The derivative of $f(x) = x$ is 1. $\frac{d}{dx}(x) = 1$ Example: The slope of the line $y = x$ is always 1.

• Power Rule: For any positive integer $n$, the derivative of $x^n$ is $nx^{n-1}$. $\frac{d}{dx}(x^n) = nx^{n-1}$ Example: If $f(x) = x^5$, then $f'(x) = 5x^4$.

• General Power Rule: For any real number $n \neq 0$, $\frac{d}{dx}(x^n) = nx^{n-1}$

Binomial Coefficient and Expansion

The binomial coefficient $\binom{n}{k}$ is used in expanding $(x+y)^n$:

• $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

• Expansion: $(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$

• Example: $\binom{5}{2} = \frac{5!}{2!3!} = 10$

II. The Constant Multiple & Sum Rules for Derivatives

These rules allow us to differentiate more complex functions by breaking them into simpler parts.

• Constant Multiple Rule: If $c$ is a constant and $f$ is differentiable, $\frac{d}{dx}(cf(x)) = c \frac{d}{dx}f(x)$ Example: If $f(x) = 3x^5$, then $f'(x) = 15x^4$.

• Sum Rule: If $f$ and $g$ are differentiable, $\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)$ Example: If $f(x) = x^5 + \sqrt{x}$, then $f'(x) = 5x^4 + \frac{1}{2}x^{-1/2}$.

Proofs and Applications

• Proofs for each rule use the definition of the derivative: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Examples demonstrate the application of these rules to polynomials and root functions.

III. The Derivative of $e^x$

The exponential function $e^x$ is unique in that its derivative is itself.

• Derivative of the Natural Exponential Function: $\frac{d}{dx}(e^x) = e^x$ Example: If $f(x) = e^x$, then $f'(x) = e^x$.

• General Exponential Functions: $\frac{d}{dx}(a^x) = a^x \ln(a)$ Example: $\frac{d}{dx}(2^x) = 2^x \ln(2)$

IV. Applications of Derivatives

Derivatives are used to analyze the behavior of functions, including finding tangent lines, rates of change, and solving real-world problems.

• Finding Tangent Lines: The tangent line to $f(x)$ at $x=a$ has slope $f'(a)$ and equation $y = f'(a)(x-a) + f(a)$.

• Horizontal Tangents: Points where $f'(x) = 0$ correspond to horizontal tangents.

• Physics Application: The position of a rock launched upward is $s(t) = 96t - 16t^2$. Velocity: $v(t) = s'(t) = 96 - 32t$ Acceleration: $a(t) = v'(t) = -32$ Example: After 1 second, $v(1) = 64$ ft/sec, $a(1) = -32$ ft/sec2.

V. Systems of Linear Equations (Contextual Example)

Finding the equation of a parabola with a given tangent line involves solving a system of equations.

• Given $y = ax^2 + bx$ and tangent at $(1,1)$ with slope $m$, set up equations: $a + b = 1$ $2a + b = m$ Solve for $a$ and $b$.

Summary Table: Derivative Rules

Function	Derivative
$c$ (constant)	$0$
$x$	$1$
$x^n$	$nx^{n-1}$
$e^x$	$e^x$
$a^x$	$a^x \ln(a)$
$cf(x)$	$c f'(x)$
$f(x) + g(x)$	$f'(x) + g'(x)$

Additional info: The notes include graphical interpretations, step-by-step proofs, and contextual applications to reinforce understanding of derivative rules and their use in calculus and physics.