Differential Calculus Exam Guide

Vocabulary-based practice flashcards covering fundamental derivative rules, transcendental function derivatives, and exam techniques for Differential Calculus.

Derivative (Fundamental Definition)

The measurement of the instantaneous rate of change or the slope of the tangent line to a curve at a given point, defined as $f'(x) = \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h}$

Constant Rule

A core algebraic rule stating that $\frac{d}{dx}(c) = 0$

Power Rule

A core algebraic rule stating that $\frac{d}{dx}(x^n) = n \cdot x^{n-1}$

Constant Multiple Rule

A core algebraic rule stating that $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$

Sum / Difference Rule

A core algebraic rule stating that $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$

Product Rule

An advanced structural rule stating that $\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)$

Quotient Rule

An advanced structural rule stating that $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$

Chain Rule

An advanced structural rule stating that $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

Derivative of $\sin(x)$

$\cos(x)$.

Derivative of $\cos(x)$

$-\sin(x)$.

Derivative of $\tan(x)$

$\sec^2(x)$.

Derivative of $\csc(x)$

$-\csc(x)\cot(x)$.

Derivative of $\sec(x)$

$\sec(x)\tan(x)$.

Derivative of $\cot(x)$

$-\csc^2(x)$.

Derivative of $e^x$

$e^x$.

Derivative of $a^x$

$a^x \ln(a)$.

Derivative of $\ln(x)$

$\frac{1}{x}$.

Derivative of $\log_a(x)$

$$\frac{1}{x \ln(a)}$$

Derivative of $\arcsin(x)$

$$\frac{1}{\sqrt{1 - x^2}}$$

Derivative of $\arccos(x)$

$\frac{-1}{\sqrt{1 - x^2}}$.

Derivative of $\arctan(x)$

$\frac{1}{1 + x^2}$.

Implicit Differentiation

A technique where both sides are differentiated with respect to $x$, treating $y$ as a function of $x$ (chain rule) which adds a $\frac{dy}{dx}$ term, then solving for $\frac{dy}{dx}$.

Logarithmic Differentiation

A technique used when variables are in the base and exponent (e.g., $x^x$) by taking the natural log ($\ln$) of both sides first to bring the exponent down, then differentiating.