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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 $$x=2π$$.

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