Calculus Derivatives and Particle Motion Notes

Practice vocabulary flashcards reviewing limit definitions of derivatives, differentiation rules (power, product, quotient, chain, trig, and implicit), and particle motion concepts based on the lecture notes.

Derivative (Definition / Meaning)

The slope at a point or the instantaneous rate of change.

Derivative as a Limit ($f'(x)$)

$$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Derivative at a Point as a Limit ($f'(a)$)

$$\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Power Rule

$$\frac{d}{dx}[x^n] = n \cdot x^{n-1}$$

Product Rule

$$\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + g'(x)f(x)$$

Quotient Rule

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

Derivative of $\sin(x)$

$$\cos(x)$$

Derivative of $\cos(x)$

$$-\sin(x)$$

Derivative of $\tan(x)$

$$\sec^2(x)$$

Derivative of $\cot(x)$

$$-\csc^2(x)$$

Derivative of $\sec(x)$

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

Derivative of $\csc(x)$

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

Chain Rule

$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

Implicit Differentiation ($\frac{d}{dx}[x^2 + y^3]$)

$$2x + 3y^2 \cdot \frac{dy}{dx}$$

Velocity ($v$)

The first derivative of position ($s$): $v = s'$

Acceleration ($a$)

The second derivative of position ($s$) and first derivative of velocity ($v$): $a = s'' = v'$

Direction of Particle

Determined by the sign of velocity ($v$).

Speed

The absolute value of velocity: $|v|$

Total Distance Traveled

Found by determining when $v = 0$, then adding all individual distances together using absolute values ($|\cdot|$).

Speed Up Condition

Occurs when velocity ($v$) and acceleration ($a$) have the same sign.

Slow Down Condition

Occurs when velocity ($v$) and acceleration ($a$) have opposite signs.