Calculus I: Derivatives, Tangents, and Rates of Change – Comprehensive Study Notes

Derivatives and Rates of Change

Average and Instantaneous Rate of Change

The average rate of change of a function over an interval measures the change in the function's value divided by the change in the input. The instantaneous rate of change is the limit of the average rate of change as the interval shrinks to a single point, and is given by the derivative.

• Average Rate of Change: $m_{avg} = \frac{f(x_1) - f(x_0)}{x_1 - x_0}$

• Instantaneous Rate of Change: $m_{tan} = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

• Example: For a rock launched vertically, the position function $s(t) = 96t - 16t^2$ gives the height above ground. The instantaneous velocity at $t=1$ is $s'(1)$, and the tangent line at $t=1$ has slope $s'(1)$.

Definition of the Derivative

The derivative of a function at a point measures the instantaneous rate of change or the slope of the tangent line at that point. It is defined as:

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

• Alternate form: $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

• Example: For $f(x) = \sqrt{x}$, compute $f'(2)$ and use it to find the tangent line at $(2, \sqrt{2})$.

Continuity and Differentiability

Relationship Between Continuity and Differentiability

A function is differentiable at a point if it is continuous there, but continuity alone does not guarantee differentiability. A function may fail to be differentiable at a point if:

• It is not continuous at that point.

• It has a corner or cusp at that point.

• It has a vertical tangent at that point.

• Example: The graph is continuous but not differentiable at $x=3$ due to a corner.

Identifying Points of Discontinuity and Non-Differentiability

• Find where the function is not continuous (e.g., jumps, holes).

• Find where the function is not differentiable (e.g., corners, vertical tangents).

• Example: For a given graph, identify intervals where $g$ is not continuous or not differentiable.

Derivative Rules and Notation

Notation for Derivatives

• $\frac{dy}{dx}$: Derivative of $y$ with respect to $x$

• $\frac{d}{dx}f(x)$: Derivative of $f(x)$ with respect to $x$

• $f'(x)$: Most common notation

• $\left.\frac{dy}{dx}\right|_{x=a}$: Derivative evaluated at $x=a$

Power Rule and Constant Multiple Rule

• Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$ for any real $n$

• Constant Multiple Rule: $\frac{d}{dx}(cf(x)) = c \frac{d}{dx}f(x)$

• Example: $\frac{d}{dx}(4x^7) = 28x^6$

Sum Rule

• $\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$

Product Rule

• $\frac{d}{dx}(f(x)g(x)) = f(x)g'(x) + f'(x)g(x)$

• Example: $\frac{d}{dx}(v^2(2\sqrt{v}+1))$

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: $\frac{d}{dx}\left(\frac{x^2+3x+4}{x+1}\right)$

Derivatives of Trigonometric Functions

• $\frac{d}{dx}(\sin x) = \cos x$

• $\frac{d}{dx}(\cos x) = -\sin x$

• $\frac{d}{dx}(\tan x) = \sec^2 x$

• $\frac{d}{dx}(\sec x) = \sec x \tan x$

• $\frac{d}{dx}(\csc x) = -\csc x \cot x$

• $\frac{d}{dx}(\cot x) = -\csc^2 x$

• Example: $\frac{d}{dx}(\sec x \csc x)$

Higher Order Derivatives

• The second derivative is $f''(x) = \frac{d^2}{dx^2}f(x)$

• The nth derivative is $f^{(n)}(x)$

• Example: Find the third derivative of $f(x) = 3x^2 - 5x + 12$

Special Derivative Techniques

Chain Rule

The chain rule is used to differentiate composite functions. If $y = f(g(x))$, then:

• $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$

• Example: $y = (5x+4)^3$, $y = \sin^2 x$, $y = \sin x^3$

General Power Rule

• If $g(x)$ is differentiable and $p$ is real, $\frac{d}{dx}(g(x))^p = p(g(x))^{p-1}g'(x)$

• Example: $\frac{d}{dx}(\tan x + 10)^3$

Implicit Differentiation

Used when $y$ is defined implicitly by an equation involving both $x$ and $y$.

• Differentiate both sides with respect to $x$, treating $y$ as a function of $x$.

• Example: $x^2 + y^2 = 1$ yields $2x + 2y \frac{dy}{dx} = 0$

• Solve for $\frac{dy}{dx}$ as needed.

Derivatives of Logarithmic and Exponential Functions

Exponential and Logarithmic Properties

• $e^{\ln x} = x$ for $x > 0$

• $\ln(e^x) = x$ for all $x$

• $\frac{d}{dx}(e^x) = e^x$

• $\frac{d}{dx}(\ln x) = \frac{1}{x}$ for $x > 0$

• General Rule: If $u$ is differentiable at $x$ and $u(x) \neq 0$, $\frac{d}{dx}(\ln|u(x)|) = \frac{u'(x)}{u(x)}$

Applications of Derivatives

Velocity and Acceleration

• Velocity: $v(t) = \frac{ds}{dt} = s'(t)$

• Acceleration: $a(t) = \frac{dv}{dt} = s''(t)$

• Example: For $s(t) = t^2 - 5t$, find $v(t)$ and $a(t)$, and analyze motion over $0 \leq t \leq 5$.

Marginal Cost and Average Cost

• Average Cost: $\text{Average Cost} = \frac{C(x)}{x}$

• Marginal Cost: $C'(x)$, the derivative of the cost function

• Example: For $C(x) = -0.02x^2 + 50x + 100$, find average and marginal cost for $x=100$ and $x=900$

Elasticity of Demand

• Elasticity: $E(x) = -\frac{x f'(x)}{f(x)}$

• Elastic if $E < -1$, inelastic if $-1 < E < 0$

• Example: For $D(p) = 286 - 20p$, compute elasticity and determine price ranges for elastic/inelastic demand.

Tables and Data Interpretation

Using Tables for Derivatives

Tables may provide values of functions and their derivatives at specific points. For example:

x	f(x)	g(x)	g'(x)
1	5	2	3
2	7	1	4

Use such tables to compute derivatives of composite functions at given points.

Summary of Key Derivative Rules

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

• Constant Multiple Rule: $\frac{d}{dx}(cf(x)) = c \frac{d}{dx}f(x)$

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

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

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

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

• Trigonometric Derivatives: $\frac{d}{dx}(\sin x) = \cos x$, $\frac{d}{dx}(\cos x) = -\sin x$, etc.

• Exponential/Logarithmic Derivatives: $\frac{d}{dx}(e^x) = e^x$, $\frac{d}{dx}(\ln x) = \frac{1}{x}$

Additional info:

• Some examples and exercises are referenced but not fully solved; students should practice by applying the rules above.

• Graphs and diagrams in the original notes illustrate concepts such as continuity, differentiability, and tangent lines.