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The derivative of a function at a point on the curve is the intantaneous rate of change of the function at that point, and is therefore the tangent of the curve at that point.


Derivatives


The derivative of a function is the instantaneous rate of change of the function with respect to one or more variable parameters. In general, if \(y=f(x)\) and the variable parameter \(x\) is given a rate of change \(Δx\), then:

\(y+Δy=f(x+Δx)\)

Then, subtracting \(y=f(x)\) to get the amount of change of the function, we get:

\(Δy=f(x+Δx)-f(x)\)

Now dividing by \(Δx\), we get the average rate of change of \(y\) with respect to \(x\) in the interval from \(x\) to \(x+Δx\), which is:

\(\frac{Δy}{Δx}=\frac{f(x+Δx)-f(x)}{Δx}\)

The derivative is the limit of \(\frac{Δy}{Δx}\) as \(Δx\) approaches zero; Or, in terms of calculus:

\(\frac{dy}{dx}=\lim\limits_{Δx\to 0}\frac{Δy}{Δx}\)or \(\frac{d}{dx}[f(x)]=\lim\limits_{Δx\to 0}\Big[\frac{f(x+Δx-f(x)}{Δx}\Big]\)

The derivative of a function is then the measure of the degree at which the curve slopes between two infinitessimally spaced points on the curve. The derivative of a curve's function at a given point on the curve is the slope or tangent of the curve at that point on the curve. We say that a function is differentiable at a point if the function is both continuous and non-asymptotic at that point. For example, a function defined such that it has an interval of \((0 \le x \le π)\) would not be differentiable at \(x\) if \(x=2π\).

Derivative
Derivative
Tangent
Tangent

While calculating the derivative of a function using the formula above will always work, elementary rules exist for quick differentiation of certain functions. In general, where \(y=x^n\):

\(\frac{dy}{dx}=n\cdot x^{n-1}=\) the derivative

Since the derivative of a function is a measure of the rate of change of the function, the derivative of a constant is zero. Also, the derivative of a variable with respect to itself is always 1. Any constant multiplier remains a multiplier in the derivative, while added constants disappear. If there are added terms, each term is differentiated separately. For example, if \(y=4x^2-\frac{x}{3}+2\), the derivative is:

\(\frac{dy}{dx}=(2)(4x)^{2-1}-\frac{x^{1-1}}{3}+0\)\(= 8x-\frac{1}{3}\)

The derivative of a function may be written in various forms. For example, \(\frac{dy}{dx}\), \(u^′(x)\), and \(Dx\), are all representative forms of the derivative. Which one to use is entirely up to style and clarity of the moment.

Derivative of a Constant


If \(f\) is a function such that \(f(x)=c\), then the derivative is zero. In symbolic terms:

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

Derivative of \(x^n\) (Power Rule)


If \(f\) is a function such that \(f(x)=x^n\), then the derivative is the expontent times \(x\) to the power \(n-1\). Note that this is true in all cases, including when \(n=0\). The power rule is generally written:

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

Derivative of a Sum (Sum Rule)


If \(u\) and \(v\) are differentiable at \(x\) and \(f\) is a function such that \(f(x)=u(x)+v(x)\), then the derivative is the sum of the derivative of \(u(x)\) and the derivative of \(v(x)\). In general:

\(\frac{d}{dx}(u\pm v)=\frac{du}{dx}\pm \frac{dv}{dx}\)\(=u^′(x)\pm v^′(x)\)

Derivative of a Product (Product Rule)


If \(u\) and \(v\) are differentiable at \(x\) and \(f\) is a function such that \(f(x)=u(x)\cdot v(x)\), then the derivative is the first factor times the derivative of the second, plus the second factor times the derivative of the first. In general:

\(\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}\)\(=u(x)v^′(x)+v(x)u^′(x)\)

Derivative of a Constant Times a Function


If \(u\) is differentiable at \(x\) and \(f\) is a function such that \(f(x)=c\cdot u(x)\), then the derivative is the constant times the derivative of the function:

\(\frac{d}{dx}(cu)=c\frac{du}{dx}\)

Derivative of a Quotient (Quotient Rule)


If \(u\) and \(v\) are differentiable at \(x\) and \(f\) is a function such that \(f(x)=\frac{u(x)}{v(x)}\), then the derivative is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

\(\frac{d}{dx}(\frac{u}{v})=\frac{v\frac{du}{dx}-\frac{dv}{dx}}{v^2}\)\(=\frac{v(x)u^′(x)-u(x)v^′(x)}{[v(x)]^2}\)

Chain Rule


If \(f\) and \(g\) are functions such that \(g\) is differntiable at \(x\) and \(f\) is differntiable at \(u=g(x)\) and \(F\) is a function such that \(F(x)=f(g(x))\), then the derivative of \(F(x)\) is the derivative \(f(u)\) times the derivative of \(g(x)\):

\(\frac{d}{dx}F(x)=\frac{d}{du}f(u)\cdot \frac{d}{dx}g(x)\)

A simple but less accurate way to state this theorem is that if \(y=f(u)\) and \(u=g(x)\), then:

\(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\)

This differential notation is meant only to be an aid to memory. Do not think of derivatives such as \(\frac{dy}{dx}\) and \(\frac{du}{dx}\) as fractions, even though they seem to behave as such.

Power Rule


If \(u\) is a function that is differentiable at \(x\) and \(f\) is a function such that \(f(x) = [u(x)]^n\), where \(n\) is a rational number, then the derivative is the exponent times \(u(x)\) to the power \(n-1\), times the derivative of \(u(x)\):

\(\frac{d}{dx}([u(x)]^n)\)\(=n[u(x)]^{n-1}\cdot\frac{d}{dx}(u(x))\)


Tables

Table of Derivatives


Calculus: Introduction Calculus: Integrals