https://tutorial.math.lamar.edu/classes/calcI/DefnOfDerivative.aspx
Defintion of the Derivative. The derivative of f (x) f ( x) with respect to x is the function f ′(x) f ′ ( x) and is defined as, f ′(x) = lim h→0 f (x+h) −f (x) h (2) (2) f ′ ( x) = lim h → 0. ⁡. f ( x + h) − f ( x) h. Note that we replaced all the a 's in (1) (1) with x 's to acknowledge the fact that the derivative is

https://www.khanacademy.org/math/differential-calculus/dc-diff-intro
The derivative of a function describes the function's instantaneous rate of change at a certain point. Another common interpretation is that the derivative gives us the slope of the line tangent to the function's graph at that point. Learn how we define the derivative using limits. Learn about a bunch of very useful rules (like the power, product, and quotient rules) that help us find

https://en.wikipedia.org/wiki/Derivative
The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function 's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear

https://math.libretexts.org/Bookshelves/Calculus/CLP-1_Differential_Calculus_(Feldman_Rechnitzer_and_Yeager)/03%3A_Derivatives/3.02%3A_Definition_of_the_Derivative
The derivative f ′ (a) at a specific point x = a, being the slope of the tangent line to the curve at x = a, and. The derivative as a function, f ′ (x) as defined in Definition 2.2.6. Of course, if we have f ′ (x) then we can always recover the derivative at a specific point by substituting x = a.

https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/03%3A_Derivatives/3.01%3A_Defining_the_Derivative
Definition: 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), is defined by. f′ (a) = lim x → a f(x) − f(a) x − a. provided this limit exists. Alternatively, we may also define the derivative of f(x) at a as.

https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/03%3A_Derivatives/3.02%3A_The_Derivative_as_a_Function
In this section, you will learn how to find the derivative of a function as a new function and how to use it to analyze the behavior of the original function. You will also see how the graphs of a function and its derivative are related. This is a key skill for calculus students and a prerequisite for the next topics. To learn more, visit the Mathematics LibreTexts website.

https://www.math.net/derivative
Derivative. The derivative of a function is the rate of change of the function's output relative to its input value. Given y = f (x), the derivative of f (x), denoted f' (x) (or df (x)/dx), is defined by the following limit: The definition of the derivative is derived from the formula for the slope of a line. Recall that the slope of a line is

https://mathworld.wolfram.com/Derivative.html
The derivative of a function represents an infinitesimal change in the function with respect to one of its variables. The "simple" derivative of a function f with respect to a variable x is denoted either f^'(x) or (df)/(dx), (1) often written in-line as df/dx. When derivatives are taken with respect to time, they are often denoted using Newton's overdot notation for fluxions, (dx)/(dt)=x^..

https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative
Learning Objectives. 3.1.1 Recognize the meaning of the tangent to a curve at a point.; 3.1.2 Calculate the slope of a tangent line.; 3.1.3 Identify the derivative as the limit of a difference quotient.; 3.1.4 Calculate the derivative of a given function at a point.; 3.1.5 Describe the velocity as a rate of change.; 3.1.6 Explain the difference between average velocity and instantaneous velocity.

https://calcworkshop.com/derivatives/definition-of-derivative/
Learn how to calculate the slope of a curve at a point using the limit definition of the derivative, which is the rate of change of a function. See examples, formulas, notation and video tutorial.

https://www.khanacademy.org/math/calculus-all-old/taking-derivatives-calc
Learn. Derivatives of sin (x), cos (x), tan (x), eˣ & ln (x) (Opens a modal) Derivative of logₐx (for any positive base a≠1) (Opens a modal) Worked example: Derivative of log₄ (x²+x) using the chain rule. (Opens a modal) Differentiating logarithmic functions using log properties. (Opens a modal)

https://www.youtube.com/watch?v=-aTLjoDT1GQ
Learn the definition of the derivative formula with this calculus video tutorial. It explains the difference quotient with limits and examples.

https://openstax.org/books/calculus-volume-1/pages/3-2-the-derivative-as-a-function
Learn how to define and graph the derivative function of a given function, and how to use the derivative to measure the rate of change or slope of a tangent line. Explore the connection between derivatives and continuity, and the conditions for a function to have a derivative.

https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new
About this unit. The derivative of a function describes the function's instantaneous rate of change at a certain point - it gives us the slope of the line tangent to the function's graph at that point. See how we define the derivative using limits, and learn to find derivatives quickly with the very useful power, product, and quotient rules.

https://www.mathwarehouse.com/calculus/derivatives/how-to-use-the-derivative-definition.php
The definition of the derivative is used to find derivatives of basic functions. Derivatives always have the $$\frac 0 0$$ indeterminate form. Consequently, we cannot evaluate directly, but have to manipulate the expression first. We can use the definition to find the derivative function, or to find the value of the derivative at a particular

https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/01%3A_Understanding_the_Derivative/1.04%3A_The_Derivative_Function
Definition 1.4.1. Let f be a function and x a value in the function's domain. We define the derivative of f, a new function called f ′, by the formula f ′ (x) = limh → 0f ( x + h) − f ( x) h, provided this limit exists. We now have two different ways of thinking about the derivative function: given a graph of y = f(x), how does this

https://math24.net/definition-derivative.html
Learn the concept of derivative as the rate of change of a function and how to find it using the limit definition. See examples of differentiating basic functions and solving problems.

https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-2/v/calculus-derivatives-1-new-hd-version
Learn how to define the derivative of a function at a point using the limit of the slope of the secant line. Watch a video, see examples, and read comments from other learners.

https://www.youtube.com/watch?v=VOIUtvAdIgs
Derivatives part 1. Definition of the derivative lecture from our introductory calculus pilot course. First of many animated videos to come from http://www.f

https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/12/MA005-3.2-Definition-of-Derivative.pdf
definition of the derivative of a function. Definition of the Derivative: The derivative of a function f is a new function, f ' (pronounced "eff prime"), whose value at x is f '(x) = 0 ( ) ( ) lim K f [ K f [o K if the limit exists and is finite. This is the definition of differential calculus, and you must know it and understand what it says.

https://en.wikipedia.org/wiki/Total_derivative
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one.In many situations, this is the same as considering all partial derivatives simultaneously.

https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-2/v/alternate-form-of-the-derivative
The derivative of a function is the measure of change in that function. Consider the parabola y=x^2. For negative x-values, on the left of the y-axis, the parabola is decreasing (falling down towards y=0), while for positive x-values, on the right of the y-axis, the parabola is increasing (shooting up from y=0).

https://www.geeksforgeeks.org/quizzes/derivatives-definition-and-basic-rules/
Derivatives: Definition and Basic Rules. The derivative of a function describes its instant rate of change at a given location. Another popular interpretation is that the derivative represents the slope of the line tangent to the function's graph at that moment. Learn how to define the derivative with limitations.

https://math.libretexts.org/Courses/Monroe_Community_College/MTH_210_Calculus_I_(Professor_Dean)/Chapter_3%3A_Derivatives/3.1%3A_Definition_of_the_Derivative
Definition. 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), is defined by. f′ (a) = lim x → af(x) − f(a) x − a. provided this limit exists. Alternatively, we may also define the derivative of f(x) at a as. f′ (a) = lim h → 0f(a + h) − f(a) h.

https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-6a/v/derivative-properties-and-polynomial-derivatives
Basic derivative rules. Let's explore how to find the derivative of any polynomial using the power rule and additional properties. The derivative of a constant is always 0, and we can pull out a scalar constant when taking the derivative. Furthermore, the derivative of a sum of two functions is simply the sum of their derivatives.

https://math.libretexts.org/Courses/Mission_College/MAT_03B_Calculus_II_(Kravets)/02%3A_Introduction_to_Differential_Equations/2.02%3A_Basics_of_Differential_Equations
Definition: differential equation. A differential equation is an equation involving an unknown function \(y=f(x)\) and one or more of its derivatives. A solution to a differential equation is a function \(y=f(x)\) that satisfies the differential equation when \(f\) and its derivatives are substituted into the equation.