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

Derivatives: Rules and Applications

General Rules for Derivatives

Derivatives measure the rate at which a function changes. The following are fundamental rules for finding derivatives:

• 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, Inverse Trigonometric, Exponential, and Logarithmic Derivatives

• 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:

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

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

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

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

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

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

• Exponential and Logarithmic Functions:

  • $\frac{d}{dx}(e^x) = e^x$

  • $\frac{d}{dx}(a^x) = a^x \ln a$

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

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

Chain Rule and Implicit 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 $g$ is differentiable at $x$ and $n$ is a real number, then:

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

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 $x$ (i.e., apply the chain rule to terms involving $y$).

• Example: For $x^2 + y^2 = 1$, differentiate both sides: $2x + 2y \frac{dy}{dx} = 0$ Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{x}{y}$

Logarithmic Differentiation

Definition and Steps

Logarithmic differentiation is useful for functions where both the base and the exponent involve variables, or for products/quotients of many functions. The steps are:

1. Take the natural logarithm of both sides: $\ln y = \ln f(x)$

2. Use properties of logarithms to simplify

3. Differentiate both sides with respect to $x$

4. Solve for $\frac{dy}{dx}$

• Example: $y = x^x$ 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

Finding the Equation of a Tangent Line

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

• $y = f(a) + f'(a)(x - a)$

To find the tangent line:

1. Find $f(a)$ and $f'(a)$

2. Substitute into the tangent line equation

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

Velocity, Speed, and Acceleration

Definitions

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

• Speed: The absolute value of velocity. $|v(t)|$

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

• Example: If $s(t) = -16t^2 + 64t + 96$, then $v(t) = -32t + 64$, $a(t) = -32$

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 time, when multiple variables are related by an equation.

1. Read the problem carefully and assign variables to all quantities that change with time.

2. Draw a diagram if possible.

3. Write an equation relating the variables.

4. Differentiate both sides with respect to time $t$ (using the chain rule as needed).

5. Substitute known values and solve for the required rate.

• Example: If a sphere's radius increases at a rate of $2$cm/s, find the rate at which its volume increases when the radius is $5$ cm. Volume of a sphere: $V = \frac{4}{3}\pi r^3$ $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$ At $r = 5$, $\frac{dr}{dt} = 2$: $\frac{dV}{dt} = 4\pi (5)^2 (2) = 200\pi$

Common Related Rates Formulas

Quantity	Formula	Derivative with respect to $t$
Circle Area	$A = \pi r^2$	$\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$
Sphere Volume	$V = \frac{4}{3}\pi r^3$	$\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$
Right Triangle (Pythagoras)	$x^2 + y^2 = r^2$	$2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 2r \frac{dr}{dt}$

Summary Table: Derivative Rules

Rule	Formula
Sum Rule	$\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$
Difference 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	$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$
Chain Rule	$\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$

Practice and Application

• Be able to apply all derivative rules to a variety of functions, including trigonometric, exponential, and logarithmic functions.

• Practice implicit and logarithmic differentiation for complex expressions.

• Use related rates techniques to solve real-world problems involving changing quantities.

• Find equations of tangent lines and interpret their meaning in context.

Additional info: Some context and examples were expanded for clarity and completeness, based on standard Calculus I curriculum.