Calculus I: Derivatives and Their Applications

Derivatives: Definitions and Interpretations

Difference Quotient and Limit Definition

The difference quotient is a fundamental concept in calculus used to approximate the slope of a function and to define the derivative. For a function $f(x)$, the difference quotient is:

• Formula: $\frac{f(x+h) - f(x)}{h}$

• As $h \to 0$, this quotient approaches the derivative of $f$ at $x$.

• Limit Definition of Derivative: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Example: For $f(x) = x^2$, $f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = 2x$

Interpretation of Derivatives

The derivative has several important interpretations in mathematics and real-world applications:

• Slope of Tangent Line: The derivative at a point gives the slope of the tangent line to the graph at that point.

• Slope of the Curve: It represents the instantaneous rate of change of the function at a specific input.

• Instantaneous Rate of Change: In applications, the derivative can represent velocity, growth rate, or other rates of change.

• Example: If $f(x)$ models position, $f'(x)$ gives velocity at $x$.

Approximation and Real-World Context

• Approximating Slope: The difference quotient can be used to estimate the slope at a point when the exact derivative is difficult to compute.

• Interpretation: In economics, the derivative can represent marginal cost; in physics, it can represent instantaneous velocity.

Rules for Differentiation

Basic Derivative Rules

• Derivative of a Linear Function: If $f(x) = mx + b$, then $f'(x) = m$

• Constant Rule: If $f(x) = b$, then $f'(x) = 0$

• Power Rule: If $f(x) = x^r$, then $f'(x) = r x^{r-1}$

• General Power Rule: $\frac{d}{dx}[g(x)]^r = r [g(x)]^{r-1} \cdot \frac{d}{dx}[g(x)]$

• Example: $f(x) = (3x+2)^4 \Rightarrow f'(x) = 4(3x+2)^3 \cdot 3$

Exponential and Logarithmic Functions

• Exponential Function: $f(x) = e^x \Rightarrow f'(x) = e^x$

• Chain Rule for Exponential: $\frac{d}{dx}(e^{g(x)}) = e^{g(x)} \cdot g'(x)$

• Natural Logarithm: $\frac{d}{dx}(\ln x) = \frac{1}{x},\ x > 0$

• Chain Rule for Logarithm: $\frac{d}{dx}[\ln g(x)] = \frac{g'(x)}{g(x)}$

• Example: $f(x) = \ln(2x+1) \Rightarrow f'(x) = \frac{2}{2x+1}$

Product and Quotient Rules

• Product Rule: $\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$

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

• Example: $f(x) = \frac{x^2}{x+1} \Rightarrow f'(x) = \frac{(x+1)\cdot 2x - x^2 \cdot 1}{(x+1)^2}$

Constant-Multiple and Sum/Difference Rules

• Constant-Multiple Rule: $\frac{d}{dx}[k f(x)] = k \cdot \frac{d}{dx}[f(x)]$

• Sum/Difference Rule: $\frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}[f(x)] \pm \frac{d}{dx}[g(x)]$

• Example: $f(x) = 3x^2 + 2x \Rightarrow f'(x) = 6x + 2$

Composite Functions and Higher Derivatives

Derivatives of Composite Functions

To differentiate composite functions, the chain rule is used:

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

• Example: $f(x) = \sin(x^2) \Rightarrow f'(x) = \cos(x^2) \cdot 2x$

First and Second Derivatives

• First Derivative: Represents the rate of change or slope.

• Second Derivative: Represents the rate of change of the rate of change (concavity, acceleration).

• Example: $f(x) = x^3 \Rightarrow f'(x) = 3x^2,\ f''(x) = 6x$

Equations of Tangent Lines

Writing Tangent Line Equations

The equation of the tangent line to $f(x)$ at $x=a$ is:

• Formula: $y = f(a) + f'(a)(x - a)$

• Example: For $f(x) = x^2$ at $x=1$, $f(1) = 1$, $f'(1) = 2$, so $y = 1 + 2(x-1)$

Partial Derivatives

Functions of Two Variables

For functions $f(x, y)$ of two variables, partial derivatives measure the rate of change with respect to one variable while holding the other constant.

• Partial Derivative with Respect to $x$: $\frac{\partial f}{\partial x}$

• Partial Derivative with Respect to $y$: $\frac{\partial f}{\partial y}$

• Example: $f(x, y) = x^2y + y^3$ $\frac{\partial f}{\partial x} = 2xy$ $\frac{\partial f}{\partial y} = x^2 + 3y^2$

• Evaluating at a Point: Substitute the values of $x$ and $y$ into the partial derivatives.

Summary Table: Derivative Rules

Rule	Formula	Example
Linear	$f(x) = mx + b \Rightarrow f'(x) = m$	$f(x) = 2x + 3 \Rightarrow f'(x) = 2$
Constant	$f(x) = b \Rightarrow f'(x) = 0$	$f(x) = 5 \Rightarrow f'(x) = 0$
Power	$f(x) = x^r \Rightarrow f'(x) = r x^{r-1}$	$f(x) = x^3 \Rightarrow f'(x) = 3x^2$
Product	$\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$	$f(x) = x^2 \cdot \sin x$
Quotient	$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$	$f(x) = \frac{x^2}{x+1}$
Chain	$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$	$f(x) = \sin(x^2)$
Exponential	$f(x) = e^x \Rightarrow f'(x) = e^x$	$f(x) = e^{2x} \Rightarrow f'(x) = 2e^{2x}$
Logarithm	$\frac{d}{dx}(\ln x) = \frac{1}{x}$	$f(x) = \ln(3x) \Rightarrow f'(x) = \frac{3}{3x} = \frac{1}{x}$

Additional info: Academic context and examples have been added to clarify each rule and application.