# Derivative **An expression measuring the rate of change of a [[function]] with respect to a change in its input.** > [!Derivative] > Let $f$ be a real-valued function defined on an *open interval* containing the point $x$. > > $f$ is *differentiable* at $x$ if the limit > $L=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$ > exists. > > The limit $L$ is the *derivative* of $f$ at $x$. The existence of a derivative of the function $f$ implies that $f$ must also be *continuous* at $x$. The derivative of a *single variable* function at a point is equal to the *slope of the tangent* to its graph at that point. >[!Explanation]+ > For *single variable* functions, the *slope* of a *secant* of their graph approximates the > *rate of change* of the function between the two points of intersection. > > If the points of intersection are at $x$ and $x+h$, then > $\text{rate of change}\approx\text{slope}_{\text{secant}}=\frac{f(x+h)-f(x)}{h}$ > Suppose $h$ *approached zero*. As $h$ decreases, the points of intersection are brought closer together. In the below diagram, this is represented by the secants increasing in opacity. > > ![[DerivativeExplanation1.svg]] > The *limit* of the slopes of the secant as $h$ approaches zero is *equal* to the slope of the *tangent* to the graph at $x$, represented by the brightest line. > > As tangents contact a curve at one point, the limit is the *best linear approximation*[^1] of the rate of change of the function at the point $x$. Thus, > $f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$ ^1860be >[!Example]- > Suppose the derivative of $f(x)=x^{3}$ is to be evaluated at $x=2$. > $\begin{align*} > f'(x)&=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} \\ > &=\lim_{h\rightarrow 0}\frac{(x+h)^{3}-x^{3}}{h} \\ > &=\lim_{h\rightarrow 0}\frac{x^{3}+3x^{2}h+3xh^{2}+h^{3}-x^{3}}{h} \\ > &=\lim_{h\rightarrow 0}3x^{2}+3xh+h^{2} \\ > &=3x^{2} \\ > \therefore f'(2)&=12 > \end{align*}$ > Shown below is the graph of $x^{3}$ and two copies of the line $12x$, with one shifted to be tangent to the graph at $x=2$. > > ![[DerivativeExample1.svg|400]] The process of determining a derivative is *differentiation* and is the inverse of [[Antiderivative|antidifferentiation]] and [[Integral|integration]]. Differentiation and integration are related by the [[fundamental theorem of calculus]]. Functions defined *implicitly* can also be differentiated through [[implicit differentiation]]. The concept of the derivative can be generalised to functions of *several real variables* with [[Partial Derivative|partial derivatives]]. ## Notation ### Leibniz's notation Leibniz's notation uses $d$ to represent *infinitesimal* increments of a variable, in a similar way to the use of $\Delta$ to represent finite increments. It is commonly used to specify the *variable of differentiation*, i.e. the independent variable. > The *first derivative* of $y=f(x)$ is denoted as > $\frac{dy}{dx}\quad\frac{df}{dx}\quad\frac{d}{dx}f$ > The *$n$th derivative* is denoted as > $\frac{d^{n}y}{dx^{n}}\quad\frac{d^{n}f}{dx^{n}}\quad\frac{d^{n}}{dx^{n}}f$ >[!Explanation]- > The notation for the $n$th derivative is an *abbreviation* of the repeated application of the *differential operator* $\frac{d}{dx}$, for example: > $\frac{d^{2}y}{dx^{2}}=\frac{d}{dx}\left(\frac{dy}{dx}\right)$ > The derivative of $y$ at the point $x=a$ is notated as > $\left.\frac{dy}{dx}\right|_{x=a}$ ### Lagrange's notation Lagrange's notation uses the *prime mark* to denote derivatives. It is used when the derivative is considered as a function itself. > The *first to third derivatives* of $f(x)$ are denoted as > $f'(x)\quad f''(x)\quad f'''(x)$ > For *larger orders* derivatives, $f^{(n)}(x)$ is used. ### Euler's notation Euler's notation uses a $D$ to explicitly symbolise the *differential operator*. > The *first and $n$th derivative* are denoted as > $Df(x)\quad D^{n}f(x)$ > A subscript may be attached to the operator to specify the *variable of differentiation*. > $D_{x}f(x)=Df(x)=f'(x)$ ## Differentiation rules The derivative of any function can be calculated using the definition, though this is often impractical. Several well-known formulas for derivatives form a set of *differentiation rules* that can be used to determine more complex derivatives. ### Constants The derivative of any constant $c\in\mathbb{R}$ is *zero*. If $f(x)=c$, then $f'(x)=0$ ### Linearity of differentiation The derivative of a *linear combination* of functions is equal to the same linear combination of their derivatives. Commonly this rule is broken down into the *constant factor* and *sum* rule. >[!Constant factor rule] > $\frac{d}{dx}a[f(x)]=a[f'(x)]$ ^eed876 >[!Sum rule] > $\frac{d}{dx}[f(x)\pm g(x)]=f'(x)\pm g'(x)$ ### Additional rules - [[Product rule]] - [[Quotient rule]] - [[Power rule]] - [[Reciprocal rule]] - [[Chain rule]] - [[Inverse function rule]] ## Extensions ### Partial derivative A [[partial derivative]] is the derivative of a *multivariable function* with respect to only *one* of its variables, while the others are *treated as constants*. $\frac{\partial}{\partial x}f\quad f_{x}\quad\partial_{x}f$ ![[PartialDerivativesExplanation1.svg|500]] *The tangent to the red line at some point $x$ is the partial derivative of the function with respect to $x$ when $y=1$.* ### Directional derivative The [[directional derivative]] of a *multivariable function* $f$ is its derivative in the direction of a vector $\mathbf{v}$ at a point $\mathbf{x}$. $\nabla_{\mathbf{V}}f(\mathbf{x})\quad f'_\mathbf{v}(\mathbf{x})\quad D_{\mathbf{v}}f(\mathbf{x})$ [^1]: Note the phrasing used. The concept of the derivative equalling the slope of the tangent leads to it often being described as an *"instantaneous rate of change"*, a confusing oxymoron since *"instantaneous"* implies a single point in time whereas *"change"* requires a range of time.