CalcRef

Topics · 24 formulas

Derivatives

Derivative rules for polynomials, trig, exponential, and inverse functions.

Limit Definition of Derivative

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

The derivative of f at x is defined as the limit of the difference quotient as h approaches 0.

Basic Rules

Open formula

Power Rule

ddx[xn]=nxn−1\frac{d}{dx}[x^n] = nx^{n-1}

Bring the exponent down as a coefficient and reduce the exponent by one. Works for any real number n.

Conditions: n can be any real number. For n = 0 the result is 0.

Basic Rules

Open formula

Constant Multiple Rule

ddx[c⋅f(x)]=c⋅f′(x)\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)

A constant factor passes through the derivative operator.

Basic Rules

Open formula

Sum/Difference Rule

ddx[f(x)±g(x)]=f′(x)±g′(x)\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)

The derivative of a sum or difference is the sum or difference of the derivatives.

Basic Rules

Open formula

Product Rule

ddx[f(x)⋅g(x)]=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)

The derivative of a product: derivative of the first times the second, plus the first times the derivative of the second.

Basic Rules

Open formula

Quotient Rule

ddx[f(x)g(x)]=f′(x)g(x)−f(x)g′(x)[g(x)]2\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

The derivative of a quotient: (derivative of top times bottom minus top times derivative of bottom) over bottom squared.

Conditions: g(x) ≠ 0

Basic Rules

Open formula

Chain Rule

ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

For composite functions: differentiate the outer function evaluated at the inner function, then multiply by the derivative of the inner function.

Basic Rules

Open formula

Derivative of sin(x)

ddx[sin⁡x]=cos⁡x\frac{d}{dx}[\sin x] = \cos x

The derivative of the sine function is the cosine function.

Trigonometric

Open formula

Derivative of cos(x)

ddx[cos⁡x]=−sin⁡x\frac{d}{dx}[\cos x] = -\sin x

The derivative of cosine is negative sine.

Trigonometric

Open formula

Derivative of tan(x)

ddx[tan⁡x]=sec⁡2x\frac{d}{dx}[\tan x] = \sec^2 x

The derivative of tangent is secant squared.

Conditions: x ≠ π/2 + nπ for integer n.

Trigonometric

Open formula

Derivative of cot(x)

ddx[cot⁡x]=−csc⁡2x\frac{d}{dx}[\cot x] = -\csc^2 x

The derivative of cotangent is negative cosecant squared.

Conditions: x ≠ nπ for integer n.

Trigonometric

Open formula

Derivative of sec(x)

ddx[sec⁡x]=sec⁡xtan⁡x\frac{d}{dx}[\sec x] = \sec x \tan x

The derivative of secant is secant times tangent.

Conditions: x ≠ π/2 + nπ for integer n.

Trigonometric

Open formula

Derivative of csc(x)

ddx[csc⁡x]=−csc⁡xcot⁡x\frac{d}{dx}[\csc x] = -\csc x \cot x

The derivative of cosecant is negative cosecant times cotangent.

Conditions: x ≠ nπ for integer n.

Trigonometric

Open formula

Derivative of eˣ

ddx[ex]=ex\frac{d}{dx}[e^x] = e^x

The exponential function eˣ is its own derivative -a unique property of the natural exponential.

Exponential & Logarithmic

Open formula

Derivative of aˣ

ddx[ax]=axln⁡a\frac{d}{dx}[a^x] = a^x \ln a

The derivative of a general exponential function aˣ is aˣ times the natural log of the base.

Conditions: a > 0 and a ≠ 1.

Exponential & Logarithmic

Open formula

Derivative of ln(x)

ddx[ln⁡x]=1x\frac{d}{dx}[\ln x] = \frac{1}{x}

The derivative of the natural logarithm is 1/x.

Conditions: x > 0.

Exponential & Logarithmic

Open formula

Derivative of log_a(x)

ddx[log⁡ax]=1xln⁡a\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

The derivative of a logarithm with base a. Use the change of base formula: log_a(x) = ln(x)/ln(a).

Conditions: x > 0, a > 0, a ≠ 1.

Exponential & Logarithmic

Open formula

Derivative of arcsin(x)

ddx[arcsin⁡x]=11−x2\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1 - x^2}}

The derivative of the inverse sine function.

Conditions: -1 < x < 1.

Inverse Trigonometric

Open formula

Derivative of arccos(x)

ddx[arccos⁡x]=−11−x2\frac{d}{dx}[\arccos x] = -\frac{1}{\sqrt{1 - x^2}}

The derivative of the inverse cosine function. Note the negative sign compared to arcsin.

Conditions: -1 < x < 1.

Inverse Trigonometric

Open formula

Derivative of arctan(x)

ddx[arctan⁡x]=11+x2\frac{d}{dx}[\arctan x] = \frac{1}{1 + x^2}

The derivative of the inverse tangent function. Valid for all real x.

Inverse Trigonometric

Open formula

Derivative of arccot(x)

ddx[arccot x]=−11+x2\frac{d}{dx}[\text{arccot}\, x] = -\frac{1}{1 + x^2}

The derivative of the inverse cotangent function. Note the negative sign compared to arctan.

Inverse Trigonometric

Open formula

Derivative of arcsec(x)

ddx[arcsec x]=1∣x∣x2−1\frac{d}{dx}[\text{arcsec}\, x] = \frac{1}{|x|\sqrt{x^2 - 1}}

The derivative of the inverse secant function.

Conditions: |x| > 1.

Inverse Trigonometric

Open formula

Derivative of arccsc(x)

ddx[arccsc x]=−1∣x∣x2−1\frac{d}{dx}[\text{arccsc}\, x] = -\frac{1}{|x|\sqrt{x^2 - 1}}

The derivative of the inverse cosecant function. Note the negative sign, mirroring the arcsec derivative.

Conditions: |x| > 1.

Inverse Trigonometric

Open formula