Limitless Calculus

Derivative Formulas

Constant Rule

$\frac{d}{dx}(c) = 0$

Power Rule

$\frac{d}{dx}(x^n) = nx^{n-1}$

Exponential Rule

$\frac{d}{dx}(e^x) = e^x$

Logarithm Rule

$\frac{d}{dx}(\ln x) = \frac{1}{x}$

Sine Rule

$\frac{d}{dx}(\sin x) = \cos x$

Cosine Rule

$\frac{d}{dx}(\cos x) = -\sin x$

Tangent Rule

$\frac{d}{dx}(\tan x) = \sec^2 x$

Product Rule

$\frac{d}{dx}(uv) = u'v + uv'$

Quotient Rule

$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$

Chain Rule

$\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$

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Integral Formulas

Constant Rule

$\int k \, dx = kx + C$

Power Rule (n ≠ -1)

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

Reciprocal Rule

$\int \frac{1}{x} \, dx = \ln|x| + C$

Exponential Rule

$\int e^x \, dx = e^x + C$

Sine Rule

$\int \sin x \, dx = -\cos x + C$

Cosine Rule

$\int \cos x \, dx = \sin x + C$

Secant Squared Rule

$\int \sec^2 x \, dx = \tan x + C$

Arctangent Rule

$\int \frac{1}{1+x^2} \, dx = \arctan x + C$

Arcsine Rule

$\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin x + C$

Integration by Parts

$\int u \, dv = uv - \int v \, du$

Substitution Rule

$\int f(g(x))g'(x) \, dx = \int f(u) \, du$