Derivatives

Definition

$f^{\prime }(x):={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

if the limit exists

• on a certain subset $C$ of domain ($f$) then we say that $f$ is differentiable on $C$
• of all elements of $D$ = Domain ($f$) then we simply say that $f$ is differentiable. In particular, $f^{\prime }$ is itself a function with: $f:D\to \mathbb{R}$

Important

$f^{\prime }(x)$ for $f(x)=x^2$ ${\lim }_{h\to 0}2x+h=2x$ $f^{\prime }(x)=2x$

Notation

Important

$\frac{df(x)}{dx}:=\frac{\partial f(x)}{\partial x}:=f^{\prime }(x):={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

$d$ … can be thought of as “difference”

Rules

constant function rule: $f(x)=c\Rightarrow f^{\prime }(x)=0$

power function rule: $f(x)=x^h\Rightarrow f^{\prime }(x)=h{x}^{h-1}+C$

multiplication by a constant: $f(x)=c\ast g(x)\Rightarrow f^{\prime }(x)=c\ast g^{\prime }(x)$

sums and differences: $f(x)=g(x)\pm h(x)\Rightarrow f^{\prime }(x)=g^{\prime }(x)\pm h^{\prime }(x)$

product rule: $f(x)=g(x)\ast h(x)\Rightarrow f^{\prime }(x)=g^{\prime }(x)\ast h(x)+g(x)\ast h^{\prime }(x)$

$$\frac{d}{dx}(x^2\cdot x^3)=2x\cdot x^3+x^2\cdot 3x^2=2x^4+3x^4=5x^4$$

quotient rule: $f(x)=\frac{g(x)}{h(x)}\Rightarrow f^{\prime }(x)=\frac{g^{\prime }(x)\ast h(x)-g(x)\ast h^{\prime }(x)}{[h(x){]}^2}$

$$\frac{d}{dx}\left(\frac{x^2}{x^3}\right)=\frac{(2x\cdot x^3-x^2\cdot 3x^2)}{(x^3{)}^2}=\frac{2x^4-3x^4}{x^6}=\frac{-x^4}{x^6}=-\frac{1}{x^2}$$

chain (composition) rule: $f(x)=g(h(x))\Rightarrow f^{\prime }(x)=g^{\prime }(h(x))\ast h^{\prime }(x)$

$$\frac{d}{dx}(x^2\circ x^3)=\frac{d}{dg}(g(x){)}^2\cdot \frac{d}{dx}(x^3)=2(g(x){)}^1\cdot 3x^2=2x^3\cdot 3x^2=6x^5$$

Info

Proof for the curious: Quotient Rule = chain + product rule $f(x)=\frac{g(x)}{h(x)}=g(x)\ast \frac{1}{h(x)}$ let $k(x)=x^{-1}$ $\frac{1}{h(x)}=k(h(x))$ $f(x)=g(x)\ast k(h(x))$ $f^{\prime }(x)=g^{\prime }(x)\ast k(h(x))+g(x)\ast k^{\prime }(h(x))$ (product rule) then $k^{\prime }(x)=-1x^{-2}$ then $k^{\prime }(h(x))=h^{\prime }(x)\ast -1h(x{)}^{-2}$ (chain rule) $f^{\prime }(x)=g^{\prime }(x)\ast k(h(x))+g(x)\ast h^{\prime }(x)\ast (-1)\ast h(x{)}^{-2}$ $f^{\prime }(x)=g^{\prime }(x)\ast k(h(x))-g(x)\ast h^{\prime }(x)\ast h(x{)}^{-2}$ $f^{\prime }(x)=\frac{g^{\prime }(x)\ast k(h(x))-g(x)\ast h^{\prime }(x)}{h(x{)}^2}$

Special functions

$\frac{d}{dx}exp(x)=exp(x)$

$\frac{d}{dx}ln=\frac{1}{x}$

$\frac{d}{dx}sin(x)=cos(x)$

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

$tan(x)...\frac{d}{dx}tan(x)=\frac{cos\ast cos-(sin\ast (-sin)}{co{s}^2}=\frac{co{s}^2+si{n}^2}{co{s}^2}=\frac{1}{co{s}^2}$

quick reminder: $a=e^{\ln a}$

Tangent Line

$t_{x_0}=f^{\prime }(x_0)(x-x_0)+f(x_0$)

Applications

• whether function is increasing/decreasing in Domain
  • $f^{\prime }(x)\ge 0$, for all $xinI$ → $f$ is increasing in $I$
  • $f^{\prime }(x)\le 0$, for all $xinI$ → $f$ is decreasing in $I$
• second-order derivative $f^{\prime \prime }(x)$
  • $f^{\prime \prime }(x)\ge 0$, for all $xinI$ → $f$ is convex in $I$
  • $f^{\prime \prime }(x)\le 0$, for all $xinI$ → $f$ is concave in $I$
  • easy to remember: $x^2$ derives twice to $2$, which is greater than $0$
• unconstrained and constrained optimization problems
• Integrals (area under the curve)
• L’Hopital (functions with either asymptote or going towards infinity)
• Differential Equations

Taylor approximation

• Taylor Approximation

Elasticity

• Elasticity