# Derivative **A derivative is the rate of change of a [[Function (in Algebra)|function]] at a particular point**, or better yet the best constant approximation of the function's rate of change at a particular point. A derivative of a line is its slope and is constant for all points. However, for curves, the rate of change changes for each point, meaning the derivative is specific to a particular point. In this case, **the derivative can be thought of as the slope of a function's tangent at a particular point.** **A derivative of $f(x)$ at $a$ is denoted as $f'(a)$, and is determined by evaluating the slope of a line connecting two points of $f(x)$ as the distance between them approaches 0.** $ f'(a) = \lim_{x \to a} \frac {f(x) - f(a)}{x-a} = \lim_{h \to 0} \frac {f(a+h) - f(a)} h $ ^zzz13y The process of differentiation is the process of finding a function's derivative. If you already have a derivative and want to reverse differentiation, you need to apply the process of [[Antiderivative|Antidifferentiation]]. Note that because differentiation ignores constant terms, antidifferentiation is not an inverse of differentiation. **The process of finding a function's maximum or minimum is essentially finding a point at which its derivative equals 0.** That represents the practical use of derivatives and why they are often described as one of most potent tools in all of calculus. To determine if a point is a maximum or a minimum you can use [[The Second Derivative Test]]. See [[The Extreme Value Theorem]] for more. **Derivatives and [[Integral|integrals]] are linked together by the [[The Fundamental Theorem of Calculus]]**, which is why many call them ‘two sides of the same coin’. A function is constantly rising across its whole domain if its first derivative is always positive. ## Derivative Notation $f'(a)$ is a derivative of $f(x)$, but it can also be written as $ f'(a) = \frac d {dx} \bigg[ f(x) \bigg] $ $\frac d {dx}$ can be read as 'take a derivative of whatever is inside the brackets with respect to $x. Sometimes $\frac {dy}{dx}$ is used instead, this only indicates that we're looking at a change of $y$ over change of $x$. It explicitly names the unit of $f$. For higher order derivatives you just repeat the prime symbol. A number in brackets is also used to indicate $n$-th derivative. Fractional notation can be used here as well $ f'''(a) = f^{(3)}(a) = \frac {d^3} {dx^3} f(x) $ ## Derivative Rules For linear functions, the tangent at any point has the same slope as the function itself, so we can say that $\frac d {dx} [kx + n] = k$. From this it follows that the derivative of any constant is 0, and that the derivative of $x$ is 1. $\frac d {dx} \bigg[c \bigg] = 0 \quad \frac d {dx} \bigg[cx\bigg] = c \quad \frac d {dx} \bigg[ c g(x) \bigg]=cg'(x)$ $\quad \text {(where $c$ is any constant)}$ The power and sum rules are enough to determine derivatives of all polynomials: ![[The Derivative Power Rule#^1osjwp]] ![[The Derivative Sum Rule#^ke50ua]] Careful with products of two functions! ![[The Derivative Product Rule#^a1ucyf]] The derivative chain rule: ![[The Derivative Chain Rule#^qnzl9i]] Which can also be expressed using Leibniz's notation: ![[The Derivative Chain Rule#^2p4rqb]] ![[Exponential Functions#^x4xm54]] ![[Logarithm Rules#^kl76b8]] ### Derived Rules ![[The Derivative Product Rule#^crr592]] ![[The Derivative Chain Rule#^1vw3jd]] ![[The Derivative Chain Rule#^h0ykgq]] $\frac d {dx} \bigg[ \frac 1 x \bigg] = - \frac 1 {x^2}$ ![[The Derivative Power Rule#^gv5xph]] ![[Rational Functions#^mmdg5g]] $\frac d {dx} \left [ \frac {jf(x) + k}{nf(x) + m} \right] = \frac {f'(x)(jm - kn)} {(nf(x)+m)^2}$ ## Derivatives of Trig Functions ![[Derivatives of Sine and Cosine#^qei2uq]] ![[Trigonometric Functions#^fc25n3]] ## Function Approximation **Because a function's derivative illustrates the function's rate of change, the function can be approximated at any point ($a$ in the following example) using the derivative.** The [[Taylor Series]] does just that. [[Newton's Method for Approximating Roots]] is an iterative method that uses derivatives and linear approximations to approximate a function's root. ## Explicit vs Implicit Differentiation Determining a derivative of a function using the above rules can be called *explicit differentiation*. However, often times (especially in physics), we need to find a derivative of an implicit curve — a curve whose points satisfy a certain property, instead of one defined with a $x \to y$ formula — say when trying to find the slope of a tangent of a circle. While determining the derivative using explicit differentiation is certainly possible, [[Implicit Differentiation]] is often easier. ## Sources - [Differentiation on Brilliant Wiki](https://brilliant.org/wiki/differentiation)