Common Derivatives

Definition

Derivatives of the standard elementary functions: exponentials, logarithms, trigonometric functions, and inverse trigonometric functions.

Intuition

These derivatives are the leaves of the differentiation tree — once known, any combination of elementary functions can be differentiated by applying the algebraic rules. The exponential $e^x$ is special because it is its own derivative; logarithm and inverse trig derivatives emerge from implicit differentiation of the defining identities.

Formal Description

Exponential and Logarithm

General exponential: For base $a>0$,

$$\frac{d}{dx}{a}^x=\underset{h\to 0}{\lim }\frac{a^{x+h}-a^x}{h}=a^x\underset{h\to 0}{\lim }\frac{a^h-1}{h}.$$

The limit $\underset{h\to 0}{\lim }\frac{a^h-1}{h}$ equals $1$ precisely when $a=e$.

Proof for $a=e$: Using $e={\lim }_{m\to 0}(1+m{)}^{1/m}$ and the binomial expansion,

$$\frac{d}{dx}{e}^x{|}_{x=0}=\underset{h\to 0}{\lim }\frac{e^h-1}{h}=\underset{h,m\to 0}{\lim }\frac{(1+m{)}^{h/m}-1}{h}=\underset{h,m\to 0}{\lim }\frac{1+h+\cdots -1}{h}=1.$$

Therefore:

$$\frac{d}{dx}{e}^x=e^x.$$

Natural logarithm: Since $\ln x$ is the inverse of $e^x$, differentiate both sides of $x=\exp (\ln x)$ with respect to $x$ and apply the chain rule:

$$1=\exp (\ln x)\cdot \frac{d}{dx}\ln x=x\cdot \frac{d}{dx}\ln x⟹\frac{d}{dx}\ln x=\frac{1}{x}.$$

For $x<0$, applying the chain rule to $\ln (-x)$:

$$\frac{d}{dx}\ln (-x)=\frac{1}{-x}\cdot (-1)=\frac{1}{x}.$$ $$\frac{d}{dx}\ln x=\frac{1}{x}(x>0),\frac{d}{dx}\ln |x|=\frac{1}{x}(x\ne 0).$$

Note: $x^{-1}=\frac{1}{x}$ is the only power $x^n$ not obtainable as the derivative of another power law (since $\frac{d}{dx}{x}^0=0$).

Trigonometric Functions

Sine and cosine are derived from the limit definition using angle-addition formulas:

$$\frac{d}{dx}\sin x=\cos x\left(\underset{h\to 0}{\lim }\frac{\sin h}{h}\right)+\sin x\left(\underset{h\to 0}{\lim }\frac{\cos h-1}{h}\right),$$ $$\frac{d}{dx}\cos x=-\sin x\left(\underset{h\to 0}{\lim }\frac{\sin h}{h}\right)+\cos x\left(\underset{h\to 0}{\lim }\frac{\cos h-1}{h}\right).$$

Using ${\lim }_{h\to 0}\frac{\sin h}{h}=1$ and ${\lim }_{h\to 0}\frac{\cos h-1}{h}=0$:

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

The remaining four functions follow from the quotient rule applied to $\tan x=\frac{\sin x}{\cos x}$, etc.:

$$\frac{d}{dx}\tan x={\sec }^{2}x,\frac{d}{dx}\cot x=-{\csc }^{2}x,$$ $$\frac{d}{dx}\sec x=\tan x\sec x,\frac{d}{dx}\csc x=-\cot x\csc x.$$

Inverse Trigonometric Functions

Derivatives are obtained via implicit differentiation and the chain rule.

Arctangent: From $\tan (\arctan x)=x$,

$$\frac{1}{{\cos }^2(\arctan x)}\cdot \frac{d}{dx}(\arctan x)=1.$$

Using $\cos (\arctan x)=\frac{1}{\sqrt{1+x^2}}$ gives:

$$\frac{d}{dx}\arctan x=\frac{1}{1+x^2}(\text{all real }x).$$

Arcsine and Arccosine: From $\sin (\arcsin x)=x$ and $\cos (\arccos x)=x$, implicit differentiation yields:

$$\frac{d}{dx}(\arcsin x)=\frac{1}{\cos (\arcsin x)},\frac{d}{dx}(\arccos x)=-\frac{1}{\sin (\arccos x)}.$$

Using $\cos (\arcsin x)=\sin (\arccos x)=\sqrt{1-x^2}$:

$$\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}},\frac{d}{dx}\arccos x=-\frac{1}{\sqrt{1-x^2}}.$$

Applications

• Differentiating any elementary function via the chain rule and algebraic rules.
• $\frac{d}{dx}{e}^x=e^x$ is fundamental to solving differential equations and defining exponential growth.
• Inverse trig derivatives appear in integration formulas (e.g., $\int \frac{dx}{1+x^2}=\arctan x+C$).

Trade-offs

• $\ln x$ is only defined (as a real function) for $x>0$; the formula $\frac{d}{dx}\ln |x|=\frac{1}{x}$ extends it to $x\ne 0$.
• Arcsine and arccosine are only defined for $-1\le x\le 1$, and their derivatives require $-1<x<1$ (strict inequality).
• The limit ${\lim }_{h\to 0}\frac{\sin h}{h}=1$ holds only when angles are measured in radians.

Links

• Differentiation Rules
• Chain Rule
• Definition of the Derivative