![[Tangent_function_animation.gif]] > See also: > - [[Limits]] > - [[Differentiation Formulas]] > - [[Implicit Differentiation]] # Differentiation *Differentiation* is the process of finding the **derivative** of a function and is the focus of differential [[Calculus]]. A function is **differentiable** if its derivative exists at each point in its domain ## Definition of Derivatives > - [[Rates of Change]] > - [[Limits]] A derivative is a function > [!info] Differentiability Implies Continuity > Let --- The following definition of a derivative combines the concept of [[rates of change]] (specifically that of [[Rates of Change#Secant and Tangent Lines|secant and tangent lines]]) with the concept of [[limits]]. > [!abstract] Limit Definition of a Derivative > Let $f(x)$ be a function defined in an open interval containing $a$. The derivative of the function $f(x)$ at $a$, denoted by $f'(a)$ (or $f$ prime), is defined by: > > $f'(a)=\lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h}$ You can only *evaluate* a derivative at a specific point. When you are finding the derivative of a function you are really finding the equation used to find the slope of the tangent line. The derivative itself does not give much immediate information about the function's behavior, however you can use the derivative to e If the derivative of a function is a constant value then it is a linear function and has a consistent slope for the entirety of the function. ### Notations of Derivatives There are several notations for derivatives If $y$ is a function of $x$, then the form $\frac{dy}{dx}$ denotes the rate of change of $y$ *relative* to the rate of change of $x$. These symbols are known as **Leibniz’s notation** (named after *Gottfried Wilhelm Leibniz*) There are several other names for a derivative - Slope of the tangent line - The instantaneous rate of change ### Higher-Order Derivatives ![[Higher Order Derivatives.png|300]] **concavity** relates to the rate of change of a function’s derivative ![[Pasted image 20240118083633.png|400]] ## Implicit Differentiation The process of taking the derivative of both sides of an equation is known as **implicit differentiation** To perform implicit differentiation on 1. **Take the derivative of both sides of the equation.** > Keep in mind that $y$, which can also be called $f(x)$, is a function of $x$. Consequently: > - If $\frac{d}{dx}(\sin x)=\cos x$ > - Then $\frac{d}{dx}(\sin y)=\cos y \frac{dy}{dx}$ > - $\frac{d}{dx}(\sin [f(x)])=\cos [f(x)] \cdot f'(x)$ > > This occurs as we must use The Chain Rule to differentiate $\sin y$ with respect to $x$. 2. Rewrite the equation so that all terms containing $\frac{dy}{dx}$ are on the left and all terms that do not contain $\frac{dy}{dx}$ are on the right. 3. Factor out $\frac{dy}{dx}$ on the left 4. Solve for $\frac{dy}{dx}$ by dividing both sides of the equation by $n$ appropriate algebraic expression ## Applications of Derivatives > See also: > - [[Derivatives and the Shape of a Graph]] ### Newton's Method Newton's Method is a technique used ch > [!abstract] Newton's Method > > $x_n = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})}$