Catalog

Worked examples.

Click any expression to load it into the calculator.

x^2

Power rule

d/dx = 2·x

Bring the exponent down, subtract 1.

x^3 - 2x + 1

Sum + power

d/dx = 3·x^2 − 2

Apply the power rule to each term, then combine.

sin(x)·cos(x)

Product rule

d/dx = cos²(x) − sin²(x)

u′v + uv′ with u = sin x, v = cos x.

x/(x+1)

Quotient rule

d/dx = 1/(x+1)²

Low-high over the square of the bottom.

ln(x^2 + 1)

Chain + log

d/dx = 2x/(x² + 1)

Chain rule: d/dx ln(u) = u′/u with u = x²+1.

e^x · sin(x)

Product + exp

d/dx = e^x·(sin x + cos x)

The derivative of e^x is e^x, so it factors out.

sqrt(1 - x^2)

Chain + sqrt

d/dx = −x / sqrt(1 − x²)

Chain rule unwraps the square root; the inner derivative contributes −2x.

sqrt(x)

Square root

d/dx = 1 / (2·sqrt(x))

Half over the square root, times the inner derivative.

sqrt(x^2 + 1)

Chain + sqrt

d/dx = x / sqrt(x² + 1)

Inner derivative is 2x, cancels with the 2 in the sqrt rule.

sqrt(sin(x))

Chain + sqrt

d/dx = cos(x) / (2·sqrt(sin(x)))

Chain rule with sin(x) inside the root.

2·sqrt(x)

Constant + sqrt

d/dx = 1 / sqrt(x)

Constant factor cancels with the 2 in the sqrt rule.

sqrt(sqrt(x))

Nested sqrt

d/dx = 1 / (4·sqrt(x)·sqrt(sqrt(x)))

Composition: derivative of x^(1/4) is (1/4)·x^(−3/4).

tan(x)

Trig identity

d/dx = sec²(x)

Equivalent to 1/cos²(x).

atan(x)

Inverse trig

d/dx = 1/(1 + x²)

One over one plus x squared.

asin(x)

Inverse trig

d/dx = 1/sqrt(1 − x²)

Defined for |x| < 1.

x^x

Generalised power

d/dx = x^x·(ln x + 1)

Differentiate u^v with both base and exponent varying.

cosh(x)

Hyperbolic

d/dx = sinh(x)

Hyperbolic cosine differentiates to hyperbolic sine.

Higher-order derivatives

The same engine handles second and third derivatives, computed by repeated application of the chain and product rules.

Function	Order	Result
x^4	2	12·x²
x^4	3	24·x
e^x	2	e^x
sin(x)	2	−sin(x)
ln(x)	2	−1/x²

Worked examples.

Higher-order derivatives

Derivative Calculator