Unit 3 Formula Sheet - MOST DERIVATIVE STUFF

Unit 2 Review

Continuity

A function $y=f(x)$ is continuous at an interior point $c$ of its domain if ${\displaystyle \underset{x\to c}{lim}=f(c)}$.

Definition of the derivative

${\displaystyle f^{\prime }(x)=\underset{n\to 0}{lim}\frac{f(x+h)-f(x)}{h}}$

Alternate definition of the derivative

$f^{\prime }$ at the point $x=a$ is ${\displaystyle \underset{x\to a}{lim}\frac{f(x)-f(a)}{x-a}}$ (provided the limit exists).

Differentiability

$y=f(x)$ is differentiable on $[a,b]$ if it has a derivative at every interior point of the interval and if the left and right handed limits are equal

Types of non-differentiability

New stuff

Basic derivative rules

Derivative of a Constant Function

If $f$ is the function with constant value $c$, then ${\displaystyle \frac{df}{dx}=\frac{d}{dx}(c)=0}$.

Power rule for Positive and Negative Integer Powers of $x$

If $n$ is a positive integer, then ${\displaystyle \frac{d}{dx}{x}^n=n{x}^{n-1}}$ (this works for negative integer powers too)

Constant Multiple Rule

If $u$ is a differentiable function of $x$ and $c$ is a constant, then ${\displaystyle \frac{d}{dx}(cu)=c\frac{du}{dx}}$.

• For example, the derivative of $3x^4$ is $3$ times the derivative of $x^4$.

Sum and Difference Rule

${\displaystyle \frac{d}{dx}(u\pm v)=\frac{du}{dx}\pm \frac{dv}{dx}}$

Product Rule

${\displaystyle \frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}}$.

Quotient Rule

At a point where $v\ne 0$, ${\displaystyle \frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}}$.

• If it helps, use the mnemonic “Low D High - High D Low”.

Economics formulas

• Average Cost: ${\displaystyle \frac{C(x)}{x}}$
• Average Increase in Cost: ${\displaystyle \frac{C(x+h)-C(x)}{h}}$
• Marginal Cost: Estimates the cost of producing one unit beyond the present level of production – ${\displaystyle \underset{h\to 0}{lim}\left(\frac{C(x+h)-C(x)}{h}\right)}$
• “actual additional cost”: $C(x+h)–C(x)$
• Demand Function: Expresses the relationship between unit price and the quantity demanded – $p=f(x)$
• Revenue: $R(x)=px$
• Marginal Revenue: $R^{\prime }(x)$
  • Estimates the increase in revenue that will result from selling one additional unit
  • “actual additional revenue made” $R(x+h)–R(x)$
• Profit: $P(x)=R(x)–C(x)$
• Marginal Profit: Measures the rate of change of the profit function P and provides us with a good approximation of the actual profit or loss realized from the sale of 1 more unit of an item sold – $P{x}^{\prime }()$
  • “actual gain in profit”: $P(x+h)–P(x)$

Particle motion formulas

• Velocity: first derivative of speed.
  • The particle stops when $v=0$.
  • The particle is moving right (or up) if the velocity is positive. The particle is moving left (or down) if the velocity is negative.
  • A particle is speeding up if the signs of velocity and acceleration are the same sign. A particle is slowing down if the signs of velocity and acceleration are opposite signs.
• Acceleration: second derivative of speed (or first derivative of velocity).
• Average acceleration
  • Equal to $\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}$ or change in velocity/change in time.
• Average velocity: displacement/travel time.
• Speed: this is just the absolute value of velocity.
• Displacement is the change in position (final position - starting position).

Trig derivatives

${\displaystyle \frac{d}{dx}\sin x=\cos x}$

${\displaystyle \frac{d}{dx}\cos x=-\sin x}$

${\displaystyle \frac{d}{dx}\tan x={\sec }^{2}x}$

${\displaystyle \frac{d}{dx}\cot x=-{\csc }^{2}x}$

${\displaystyle \frac{d}{dx}\sec x=\sec x\tan x}$

${\displaystyle \frac{d}{dx}\csc x=-\csc x\cot x}$

The chain rule

If $f$ is differentiable at the point $u=g(x)$, and $g$ is differentiable at
$x$, then the composite function $(f\circ g)(x)=f(g(x))$ is differentiable at $x$, and $(f\circ g{)}^{\prime }(x)=f^{\prime }(g(x))\cdot g^{\prime }(x)$.

Implicit Differentiation

• Basically, just chain $y$ with its derivative with respect to $x$, which is ${\displaystyle \frac{dy}{dx}}$, whenever you see it.

Inverse trig derivatives

You can just derive these, so don’t put too much effort into memorizing them.

• The derivative of arcsin x is ${\displaystyle d/dx(\arcsin x)=1/\sqrt{1-x²}}$, when $-1<x<1$
• The derivative of arccos x is ${\displaystyle d/dx(\arccos x)=-1/\sqrt{1-x²}}$, when $-1<x<1$
• The derivative of arctan x is ${\displaystyle d/dx(\arctan x)=1/(1+x²)}$, for all $x$ in $R$
• The derivative of arccsc x is ${\displaystyle d/dx(cs{c}^{-1}x)=-1/(|x|\sqrt{x²-1})}$, when $x<-1$ or $x>1$
• The derivative of arcsec x is ${\displaystyle d/dx(se{c}^{-1}x)=1/(|x|\sqrt{x²-1}}$), when $x<-1$ or $x>1$
• The derivative of arccot x is ${\displaystyle d/dx(co{t}^{-1}x)=-1/(1+x²)}$, for all $x$ in $R$

Exponent and log formulas

Derivative of $e^x$

• The derivative of $e^x$ is simply $e^x$.

Derivative of any general $e^u$

${\displaystyle \frac{d}{dx}(e^u)=e^u\left(\frac{du}{dx}\right)}$

Derivative of any general $a^u$

${\displaystyle \frac{d}{dx}(a^u)=a^u\ln a\left(\frac{du}{dx}\right)}$

Derivative of $\ln x$

• Simply ${\displaystyle \frac{1}{x}}$.

Derivative of any general $\ln u$

${\displaystyle \frac{d}{dx}(\ln u)=\frac{1}{u}\left(\frac{du}{dx}\right)}$

Derivative of ${\log }_{a}x$

${\log }_{a}x$ is ${\displaystyle \frac{1}{x\ln a}}$.

Any general ${\log }_{a}u$

${\displaystyle \frac{d}{dx}({\log }_{a}u)=\frac{1}{u\ln a}\left(\frac{du}{dx}\right)}$