What are common derivatives?

Common derivatives are the derivative identities that come up most often in calculus: the derivatives of polynomials, exponentials, logarithms, the six trig functions, the six inverse trig functions, and their hyperbolic counterparts. Together they cover the building blocks needed for almost any differentiation problem when combined with the linearity, product, quotient, and chain rules.

What is the derivative of sin x and cos x?

The derivative of sin(x) is cos(x). The derivative of cos(x) is negative sin(x). The sign change on cosine reflects the 90-degree phase shift between the two functions. Repeatedly differentiating cycles through sin, cos, negative sin, negative cos, and back to sin every four steps.

What is the power rule?

The power rule says that the derivative of x to the n is n times x to the (n minus 1) for any real exponent n. To use it, multiply by the exponent and then subtract one from the exponent. The rule covers polynomials, roots like the square root of x, and reciprocals like 1 over x.

What is the difference between the product rule and the chain rule?

The product rule differentiates a product of two functions: the derivative of f times g is f-prime times g plus f times g-prime. The chain rule differentiates a composition: the derivative of f of g(x) is f-prime evaluated at g(x), times g-prime of x. Products combine functions side by side; compositions nest one function inside another.

How do I memorize derivative rules?

Group the identities by category: polynomials, exponentials and logarithms, trig, inverse trig, hyperbolic, and inverse hyperbolic. Notice the patterns: cosine differentiates to negative sine but cosh differentiates to plus sinh; arcsin and arccos have opposite signs because they sum to a constant. Drill with the drag puzzle on this page to lock the pairings in.

Common Derivatives

Reference table of the most-used derivative identities. Try puzzle mode to drill, or read the full derivatives explanation →

Derivative Tool

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Polynomial

\frac{d}{dx}\!\left[c\right]

↓

0

Polynomial

\frac{d}{dx}\!\left[x\right]

↓

1

Polynomial

\frac{d}{dx}\!\left[x^n\right]

↓

nx^{n-1}

Polynomial

\frac{d}{dx}\!\left[\frac{1}{x}\right]

↓

-\frac{1}{x^2}

Polynomial

\frac{d}{dx}\!\left[\sqrt{x}\right]

↓

\frac{1}{2\sqrt{x}}

Exp / Log

\frac{d}{dx}\!\left[e^x\right]

↓

e^x

Exp / Log

\frac{d}{dx}\!\left[a^x\right]

↓

a^x \ln(a)

Exp / Log

\frac{d}{dx}\!\left[\ln(x)\right]

↓

\frac{1}{x}

Exp / Log

\frac{d}{dx}\!\left[\log_a(x)\right]

↓

\frac{1}{x \ln(a)}

Trig

\frac{d}{dx}\!\left[\sin(x)\right]

↓

\cos(x)

Trig

\frac{d}{dx}\!\left[\cos(x)\right]

↓

-\sin(x)

Trig

\frac{d}{dx}\!\left[\tan(x)\right]

↓

\sec^2(x)

Trig

\frac{d}{dx}\!\left[\cot(x)\right]

↓

-\csc^2(x)

Trig

\frac{d}{dx}\!\left[\sec(x)\right]

↓

\sec(x)\tan(x)

Trig

\frac{d}{dx}\!\left[\csc(x)\right]

↓

-\csc(x)\cot(x)

Inverse trig

\frac{d}{dx}\!\left[\arcsin(x)\right]

↓

\frac{1}{\sqrt{1-x^2}}

Inverse trig

\frac{d}{dx}\!\left[\arccos(x)\right]

↓

-\frac{1}{\sqrt{1-x^2}}

Inverse trig

\frac{d}{dx}\!\left[\arctan(x)\right]

↓

\frac{1}{1+x^2}

Inverse trig

\frac{d}{dx}\!\left[\operatorname{arccot}(x)\right]

↓

-\frac{1}{1+x^2}

Inverse trig

\frac{d}{dx}\!\left[\operatorname{arcsec}(x)\right]

↓

\frac{1}{|x|\sqrt{x^2-1}}

Inverse trig

\frac{d}{dx}\!\left[\operatorname{arccsc}(x)\right]

↓

-\frac{1}{|x|\sqrt{x^2-1}}

Hyperbolic

\frac{d}{dx}\!\left[\sinh(x)\right]

↓

\cosh(x)

Hyperbolic

\frac{d}{dx}\!\left[\cosh(x)\right]

↓

\sinh(x)

Hyperbolic

\frac{d}{dx}\!\left[\tanh(x)\right]

↓

\operatorname{sech}^2(x)

Hyperbolic

\frac{d}{dx}\!\left[\coth(x)\right]

↓

-\operatorname{csch}^2(x)

Hyperbolic

\frac{d}{dx}\!\left[\operatorname{sech}(x)\right]

↓

-\operatorname{sech}(x)\tanh(x)

Hyperbolic

\frac{d}{dx}\!\left[\operatorname{csch}(x)\right]

↓

-\operatorname{csch}(x)\coth(x)

Inverse hyp.

\frac{d}{dx}\!\left[\operatorname{arcsinh}(x)\right]

↓

\frac{1}{\sqrt{x^2+1}}

Inverse hyp.

\frac{d}{dx}\!\left[\operatorname{arccosh}(x)\right]

↓

\frac{1}{\sqrt{x^2-1}}

Inverse hyp.

\frac{d}{dx}\!\left[\operatorname{arctanh}(x)\right]

↓

\frac{1}{1-x^2}

Differentiation rules

The four structural rules that combine and extend the identities above.

⊕

Linearity

The derivative of a sum is the sum of derivatives. Constants pull out: $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$ .

\frac{d}{dx}[3x^2 + 5x] = 6x + 5

Product rule

$\frac{d}{dx}[fg] = f'g + fg'$ . Differentiate one factor, leave the other alone, then sum.

\frac{d}{dx}[x \sin(x)] = \sin(x) + x\cos(x)

Quotient rule

$\frac{d}{dx}\!\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}$ . Mnemonic: low d-high minus high d-low, over low squared.

\frac{d}{dx}\!\left[\frac{\sin(x)}{x}\right] = \frac{x\cos(x) - \sin(x)}{x^2}

∘

Chain rule

$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$ . Differentiate the outside, then multiply by the derivative of the inside.

\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x

Common Derivatives

Derivative Tool

Categories

Polynomial

Exponential & log

Trigonometric

Inverse trig

Hyperbolic

Inverse hyperbolic

Reciprocal forms

Differentiation rules

Linearity

Product rule

Quotient rule

Chain rule