Calculus Exam 2 Study Guide: Derivatives, Implicit Differentiation, Tangent Lines, Logarithmic Differentiation, and Related Rates

Derivatives: Rules and Applications

General Rules for Derivatives

Derivatives measure the rate at which a function changes. The following are key rules for finding derivatives of various types of functions:

• Constant Rule: The derivative of a constant is zero. $\frac{d}{dx}(c) = 0$

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

• Sum Rule: The derivative of a sum is the sum of the derivatives. $\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)$

• Difference Rule: The derivative of a difference is the difference of the derivatives. $\frac{d}{dx}(f(x) - g(x)) = f'(x) - g'(x)$

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

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

Trigonometric Functions

• $\frac{d}{dx}(\sin x) = \cos x$

• $\frac{d}{dx}(\cos x) = -\sin x$

• $\frac{d}{dx}(\tan x) = \sec^2 x$

• $\frac{d}{dx}(\cot x) = -\csc^2 x$

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

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

Inverse Trigonometric Functions

• $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1 - x^2}}$

• $\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1 - x^2}}$

• $\frac{d}{dx}(\arctan x) = \frac{1}{1 + x^2}$

• $\frac{d}{dx}(\arccot x) = -\frac{1}{1 + x^2}$

Exponential and Logarithmic Functions

• $\frac{d}{dx}(e^x) = e^x$

• $\frac{d}{dx}(a^x) = a^x \ln a$

• $\frac{d}{dx}(\ln x) = \frac{1}{x}$

• $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$

Techniques of Differentiation

Chain Rule

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

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

Theorem 3.14 (Chain Rule for Powers): If $f$ is differentiable at $x$ and $n$ is a real number, then:

• $\frac{d}{dx}[f(x)]^n = n[f(x)]^{n-1} f'(x)$

Implicit Differentiation

Implicit differentiation is used when it is difficult or impossible to solve for $y$ explicitly in terms of $x$. Differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$.

• Example: For $x^2 + y^2 = 3x^2y$, differentiate both sides: $2x + 2y \frac{dy}{dx} = 6xy + 3x^2 \frac{dy}{dx}$ Rearranging and solving for $\frac{dy}{dx}$ gives: $\frac{dy}{dx} = \frac{6xy - 2x}{2y - 3x^2}$

Logarithmic Differentiation

Logarithmic differentiation is useful for functions involving products, quotients, or powers where both the base and exponent are functions of $x$.

• Take the natural logarithm of both sides, then differentiate using the chain rule.

• Example: For $y = x^x$, take $\ln$ of both sides: $\ln y = x \ln x$ Differentiate: $\frac{1}{y} \frac{dy}{dx} = \ln x + 1$ So $\frac{dy}{dx} = y(\ln x + 1) = x^x(\ln x + 1)$

Tangent Lines and Applications

Finding the Equation of a Tangent Line

The tangent line to the curve $y = f(x)$ at $x = a$ has the equation:

• $y = f(a) + f'(a)(x - a)$

• Example: For $y = x^2$ at $x = 2$: $f(2) = 4$, $f'(2) = 4$ Tangent line: $y = 4 + 4(x - 2)$

Velocity, Speed, and Acceleration

Definitions

• Velocity: The derivative of position with respect to time. $v = \frac{ds}{dt} = s'(t)$

• Speed: The absolute value of velocity. $|v| = |s'(t)|$

• Acceleration: The derivative of velocity with respect to time. $a = \frac{dv}{dt} = s''(t)$

Example: Motion in a Gravitational Field

• Given $s(t) = -16t^2 + 64t + 96$, find velocity and acceleration: $v(t) = \frac{ds}{dt} = -32t + 64$ $a(t) = \frac{dv}{dt} = -32$

• Maximum height occurs when $v(t) = 0$: $-32t + 64 = 0 \implies t = 2$ $s(2) = -16(2)^2 + 64(2) + 96 = 160$

Related Rates

Procedure for Solving Related Rates Problems

Related rates problems involve finding the rate at which one quantity changes with respect to another, often using implicit differentiation.

• Read the problem carefully and assign variables to all quantities.

• Draw a diagram if applicable.

• Write an equation relating the variables.

• Differentiating both sides with respect to time $t$.

• Substitute known values and solve for the desired rate.

Example: Spreading Oil Spill

• If oil spreads in a circle and the radius increases at a known rate, relate area $A$ and radius $r$: $A = \pi r^2$ $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$

Example: Converging Airplanes

• Use the Pythagorean theorem to relate distances and rates of change.

• If $x$ and $y$ are distances, and $z$ is the distance between planes: $z^2 = x^2 + y^2$ $2z \frac{dz}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt}$ $\frac{dz}{dt} = \frac{x \frac{dx}{dt} + y \frac{dy}{dt}}{z}$

Summary Table: Derivative Rules

Function	Derivative
$c$ (constant)	$0$
$x^n$	$n x^{n-1}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$e^x$	$e^x$
$\ln x$	$\frac{1}{x}$
$f(x)g(x)$	$f'(x)g(x) + f(x)g'(x)$
$\frac{f(x)}{g(x)}$	$\frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

Practice and Application

• Be able to apply all derivative rules to polynomial, trigonometric, exponential, and logarithmic functions.

• Use implicit and logarithmic differentiation for complex functions.

• Find equations of tangent lines and solve related rates problems using differentiation.

• Interpret velocity, speed, and acceleration in the context of motion problems.

Additional info: The notes cover all major topics for a college Calculus exam, including rules for derivatives, implicit and logarithmic differentiation, tangent lines, velocity/acceleration, and related rates. Examples and tables are provided for clarity and application.