Calculus Derivative Rules Sheet
Choose a rule from the outer structure of the expression.
Preserve inner expressions and copy untouched factors carefully.
Check signs, domains, and whether a chain-rule factor is missing.
This derivative rules sheet collects the formulas most often used in first-semester calculus. It works best as a decision guide: identify whether the expression is a sum, product, quotient, or composition, then select the corresponding rule. A formula is useful only when its conditions and notation are understood, so each group below includes a quick recognition cue. Print the page for review, then practice applying each rule without looking at the sheet.
Linearity and powers
\[\frac{d}{dx}[c]=0,\qquad \frac{d}{dx}[cf(x)]=cf'(x)\]
\[\frac{d}{dx}[f(x)+g(x)]=f'(x)+g'(x)\]
\[\frac{d}{dx}[x^n]=nx^{n-1}\]
The power rule applies to positive, zero, negative, and fractional powers wherever the original real-valued expression and derivative are defined.
Products, quotients, and compositions
\[(fg)'=f'g+fg'\]
\[\left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2},\qquad g\ne0\]
\[\frac{d}{dx}f(g(x))=f'(g(x))g'(x)\]
Use the product rule when separate variable expressions are multiplied, the quotient rule when one variable expression divides another, and the chain rule when one function is nested inside another.
Exponential and logarithmic derivatives
\[\frac{d}{dx}e^x=e^x,\qquad \frac{d}{dx}a^x=a^x\ln a\]
\[\frac{d}{dx}\ln x=\frac1x,\qquad \frac{d}{dx}\log_{10}x=\frac{1}{x\ln10}\]
For a composite exponent or logarithm, include the chain rule: \((e^{g(x)})'=e^{g(x)}g'(x)\) and \((\ln g(x))'=g'(x)/g(x)\) where \(g(x)>0\).
Trigonometric derivatives
\[(\sin x)'=\cos x,\quad (\cos x)'=-\sin x,\quad (\tan x)'=\sec^2x\]
\[(\sec x)'=\sec x\tan x,\quad (\csc x)'=-\csc x\cot x,\quad (\cot x)'=-\csc^2x\]
These formulas assume angles measured in radians. With an inner function \(g(x)\), multiply every result by \(g'(x)\).
Inverse trigonometric derivatives
\[(\arcsin x)'=\frac1{\sqrt{1-x^2}},\qquad (\arctan x)'=\frac1{1+x^2}\]
The arcsine formula applies for \(-1<x<1\). Its derivative becomes unbounded at the endpoints. The arctangent derivative is defined for every real number.
A fast rule-selection checklist
- Simplify obvious algebra first, especially a single monomial denominator.
- Identify the outermost operation: addition, multiplication, division, or composition.
- Write the matching rule before substituting the functions.
- Apply the chain rule to every nontrivial inner function.
- Simplify only after the full derivative is written.
- Carry forward restrictions from roots, logarithms, and denominators.
Frequent exam errors
- Using \((fg)'=f'g'\) instead of the two-term product rule.
- Reversing the quotient-rule numerator and losing its sign.
- Forgetting the derivative of the inside function in a composite.
- Treating \(e^x\) like a power and reducing its exponent.
- Using degree-mode intuition with derivative formulas that assume radians.
When a formula needs more context, open the full lessons on the chain rule, product and quotient rules, or trigonometric derivatives. The worked solutions library shows the same rules applied to individual expressions.
Reference standard: Rule statements and notation follow OpenStax Calculus Volume 1, Chapter 3. DerivCalc's explanations and examples are independently written.