Calculus Study Guide: Differentiation, Tangent Lines, and Applications

Differentiation: Definitions and Techniques

Definition of the Derivative

The derivative of a function at a point measures the rate at which the function value changes as its input changes. It is defined as:

• Definition: The derivative of f at a is given by $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Interpretation: The derivative represents the slope of the tangent line to the graph of f at x = a.

• Example: For $f(x) = \sqrt{3x+1}$ at $x=8$, compute $f'(8)$ using the definition above.

Equation of the Tangent Line

The tangent line to the graph of f at the point $(a, f(a))$ has the equation:

• $y = f'(a)(x - a) + f(a)$

• Example: Find the tangent line to $f(x) = \sqrt{x^2 + 1}$ at $x = 7$.

Graphical Interpretation of Derivatives

Piecewise Linear Functions and Their Derivatives

When a function is composed of line segments, its derivative is constant on each segment and may be undefined at the points where segments meet.

• Key Point: The derivative of a piecewise linear function is a step function, with jumps at the points where the slope changes.

• Example: Given a graph made of line segments, sketch the derivative by assigning the slope of each segment as the value of the derivative on that interval.

Step Functions and Their Derivatives

A step function is constant on intervals and jumps at certain points. Its derivative is zero everywhere except at the jump points, where it is undefined.

• Key Point: The derivative of a step function is zero except at discontinuities.

Graph Matching: Functions and Their Derivatives

Identifying Derivatives from Graphs

Given a graph of a function, you may be asked to select the graph of its derivative from several options.

• Key Point: The slope of the tangent to the function at each point corresponds to the value of the derivative at that point.

• Example: If the function is increasing, its derivative is positive; if decreasing, negative; if constant, zero.

Continuity and Differentiability

Definitions

• Continuous Function: A function is continuous at a point if its graph has no breaks at that point.

• Differentiable Function: A function is differentiable at a point if its derivative exists there.

• Key Point: Differentiability implies continuity, but continuity does not imply differentiability.

• Example: The function $f(x) = |x|$ is continuous everywhere but not differentiable at $x = 0$.

Derivative Rules and Applications

Basic Derivative Rules

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

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

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

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

Derivatives of Common Functions

• $\frac{d}{dx} \sin x = \cos x$

• $\frac{d}{dx} \cos x = -\sin x$

• $\frac{d}{dx} e^x = e^x$

• $\frac{d}{dx} \ln x = \frac{1}{x}$

Higher Order Derivatives

• Second Derivative: $f''(x) = \frac{d^2}{dx^2} f(x)$

• Third Derivative: $f'''(x) = \frac{d^3}{dx^3} f(x)$

Implicit Differentiation

Used when a function is defined implicitly rather than explicitly.

• Example: For $y^2 + x^2 = 1$, differentiate both sides with respect to $x$ to find $\frac{dy}{dx}$.

Applications of Derivatives

Tangent Lines

• Equation: $y = f'(a)(x - a) + f(a)$

• Application: Used to approximate function values near $x = a$.

Velocity and Acceleration

• Velocity: The derivative of position with respect to time: $v(t) = s'(t)$

• Acceleration: The derivative of velocity: $a(t) = v'(t) = s''(t)$

• Example: If $s(t) = t^2$, then $v(t) = 2t$, $a(t) = 2$.

Motion Problems

• Displacement: Change in position over a time interval.

• Average Velocity: $\frac{s(t_2) - s(t_1)}{t_2 - t_1}$

• Example: For $s(t) = -16t^2 + v_0 t + s_0$, find when the stone hits the ground by solving $s(t) = 0$.

Table: Derivative Rules Summary

Rule	Formula	Example
Power Rule	$\frac{d}{dx} x^n = n x^{n-1}$	$\frac{d}{dx} x^3 = 3x^2$
Product Rule	$\frac{d}{dx}[u v] = u' v + u v'$	$\frac{d}{dx}(x^2 \sin x) = 2x \sin x + x^2 \cos x$
Quotient Rule	$\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}$	$\frac{d}{dx}\left[\frac{x}{\sin x}\right] = \frac{1 \cdot \sin x - x \cdot \cos x}{\sin^2 x}$
Chain Rule	$\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$	$\frac{d}{dx} \sin(x^2) = \cos(x^2) \cdot 2x$

Short Answer and Multiple Choice Practice

Practice Problems

• Find $\frac{dy}{dx}$ for $y = (\cos(2x))^4$

• Find the equation of the tangent line for $y = e^x$ at $x = 3$

• Find the body's acceleration at $t = \pi/3$ sec for $s = \sin t - \cos t$

• Given $y = \tan(e^{\sqrt{x}})$, find $\frac{dy}{dx}$

Additional info:

• Some problems involve matching graphs of functions to their derivatives, which is a common calculus skill.

• Motion problems use derivatives to analyze velocity and acceleration, key applications in physics.

• Implicit differentiation and higher order derivatives are included, indicating coverage of core calculus topics.