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)

    • \frac{d}{dx}\log_{4}x=\frac{1}{x\ln 4}

  • Example 4: Differentiate f(x)=\arcsin(\tfrac{x}{2})

    • Inner function derivative: \frac{d}{dx}\left(\frac{x}{2}\right)=\frac{1}{2}

    • Outer derivative: \frac{d}{dx}\arcsin(u)=\frac{u'}{\sqrt{1-u^{2}}}

    • 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}}

Connections to foundational principles

  • Derivative rules are built from limit definitions and linearity: constants, sums, products, quotients, chain rule.

  • These standard derivatives form the core toolkit for differentiating composite functions, solving optimization, and finding rates of change in real-world problems.

  • 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.

Practical implications

  • Domain restrictions are essential: misapplying a derivative outside its domain (e.g., ln x for x≤0) leads to incorrect results.

  • When using logs with non-standard bases, convert to natural log: (\log_{a}x = \dfrac{\ln x}{\ln a}) to compute derivatives consistently.

  • Inverse trigonometric derivatives often require attention to principal branches; remember the standard ranges for each inverse function.

Quick summary to memorize

  • Power and exponential: \frac{d}{dx}x^{n}=nx^{n-1},\quad \frac{d}{dx}a^{x}=a^{x}\ln a

  • 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

  • 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

  • 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.