Common Derivatives

Elementary Derivatives: Basic Rules

• d/dx of a power with variable exponent: $\frac{d}{dx}x^{n}=nx^{n-1}$

• d/dx of an exponential with base a: $\frac{d}{dx}a^{x}=a^{x}\ln a$

• Special case with base e: $\frac{d}{dx}e^{x}=e^{x}$

• Constant rule: $\frac{d}{dx}c=0\quad\text{(where }c\text{ is constant)}$

Derivatives of basic trigonometric functions

• $\frac{d}{dx}\sin x=\cos x$

• $\frac{d}{dx}\cos x=-\sin x$

• $\frac{d}{dx}\tan x=\sec^{2}x$

• $\frac{d}{dx}\cot x=-\csc^{2}x$

• $\frac{d}{dx}\sec x=\sec x\tan x$

• $\frac{d}{dx}\csc x=-\csc x\cot x$

Derivatives of natural and general logarithmic functions

• \frac{d}{dx}\ln x=\frac{1}{x}\quad(x>0)

• $\frac{d}{dx}\ln|x|=\frac{1}{x}\quad(x\neq 0)$

• \frac{d}{dx}\log_{a}x=\frac{1}{x\ln a}\quad(a>0,\ a\neq 1)

Derivatives of inverse trigonometric functions

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

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

• \frac{d}{dx}\arctan x=\frac{1}{1+x^{2}}\quad(-\infty<x<\infty)

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

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

Notes on domains and notation

• For ln x: domain is x>0

• For ln|x|: domain is $x\neq 0$

• For arcsin/arccos: domain is $-1\le x\le 1$

• For arctan: domain is all real numbers

• For arcsec/arccsc: domain is |x|>1

• For log base a of x: base a must be positive and not equal to 1, i.e., a>0, a\neq 1

Quick reference: table of common derivatives

• Power: $\frac{d}{dx}x^{n}=nx^{n-1}$

• Exponential: \frac{d}{dx}a^{x}=a^{x}\ln a\quad(a>0)

• Natural exponential: $\frac{d}{dx}e^{x}=e^{x}$

• Sine/cosine: $\frac{d}{dx}\sin x=\cos x\,,\quad \frac{d}{dx}\cos x=-\sin x$

• Tangent/cotangent: $\frac{d}{dx}\tan x=\sec^{2}x\,,\quad \frac{d}{dx}\cot x=-\csc^{2}x$

• Secant/cosecant: $\frac{d}{dx}\sec x=\sec x\tan x\,,\quad \frac{d}{dx}\csc x=-\csc x\cot x$

• Natural log: \frac{d}{dx}\ln x=\frac{1}{x}\,(x>0)

• Logarithm with base a: $\frac{d}{dx}\log_{a}x=\frac{1}{x\ln a}$

• Inverse trig: arcsin, arccos, arctan as shown above with their domains

• Inverse secant/cosecant: arcsec/arccsc as shown above

Worked examples (with chain rule context)

• Example 1: Differentiate f(x)=\sin(3x)

  • Using chain rule: $\frac{d}{dx}\sin(3x)=\cos(3x)\cdot 3=3\cos(3x)$

• Example 2: Differentiate f(x)=e^{2x}

  • $\frac{d}{dx}e^{2x}=e^{2x}\cdot 2=2e^{2x}$

• Example 3: Differentiate f(x)=\log_{4}(x)$</p><ul><li><p>$\frac{d}{dx}\log_{4}x=\frac{1}{x\ln 4}$</p></li></ul></li><li><p>Example 4: Differentiate f(x)=\arcsin(\tfrac{x}{2})</p><ul><li><p>Inner function derivative:$\frac{d}{dx}\left(\frac{x}{2}\right)=\frac{1}{2}$</p></li><li><p>Outer derivative:$\frac{d}{dx}\arcsin(u)=\frac{u'}{\sqrt{1-u^{2}}}$</p></li><li><p>Result:$\frac{d}{dx}\arcsin\left(\frac{x}{2}\right)=\frac{\frac{1}{2}}{\sqrt{1-(x/2)^{2}}}=\frac{1}{2\sqrt{1-x^{2}/4}}$</p></li></ul></li></ul><h4 id="05b689ba-f6d8-4c13-acee-b7f6abc32516" data-toc-id="05b689ba-f6d8-4c13-acee-b7f6abc32516" collapsed="false" seolevelmigrated="true">Connections to foundational principles</h4><ul><li><p>Derivative rules are built from limit definitions and linearity: constants, sums, products, quotients, chain rule.</p></li><li><p>These standard derivatives form the core toolkit for differentiating composite functions, solving optimization, and finding rates of change in real-world problems.</p></li><li><p>The inverse function derivatives tie to the idea that derivatives of inverse functions are reciprocals of the derivatives of the original function at corresponding points.</p></li></ul><h4 id="ad38b72e-f626-45a2-bd3b-a51531639d6a" data-toc-id="ad38b72e-f626-45a2-bd3b-a51531639d6a" collapsed="false" seolevelmigrated="true">Practical implications</h4><ul><li><p>Domain restrictions are essential: misapplying a derivative outside its domain (e.g., ln x for x≤0) leads to incorrect results.</p></li><li><p>When using logs with non-standard bases, convert to natural log: (\log_{a}x = \dfrac{\ln x}{\ln a}) to compute derivatives consistently.</p></li><li><p>Inverse trigonometric derivatives often require attention to principal branches; remember the standard ranges for each inverse function.</p></li></ul><h4 id="d825676c-f9cd-412f-a759-d9ce2f7785b2" data-toc-id="d825676c-f9cd-412f-a759-d9ce2f7785b2" collapsed="false" seolevelmigrated="true">Quick summary to memorize</h4><ul><li><p>Power and exponential:$\frac{d}{dx}x^{n}=nx^{n-1},\quad \frac{d}{dx}a^{x}=a^{x}\ln a$</p></li><li><p>Trig basic:$\frac{d}{dx}\sin x=\cos x,\quad \frac{d}{dx}\cos x=-\sin x,\quad \frac{d}{dx}\tan x=\sec^{2}x$</p></li><li><p>Reciprocal trig:$\frac{d}{dx}\sec x=\sec x\tan x,
  \frac{d}{dx}\csc x=-\csc x\cot x,
  \frac{d}{dx}\cot x=-\csc^{2}x$</p></li><li><p>Logarithms:$\frac{d}{dx}\ln x=\frac{1}{x},\quad \frac{d}{dx}\log_{a}x=\frac{1}{x\ln a}$$

• Inverse trig: arcsin, arccos, arctan derivatives as listed above with their domain constraints.