# What is Calculus derivative: Definition and 18 Discussions

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
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 approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.
Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.
The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.

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1. ### Teaching Differential Calculus as the Limit of Discrete Calculus

I'm teaching Calc I. this semester and we're now covering the derivatives of power function and exponential functions as well as the basic rules, e.g. linearity and product rule. Some years back I ran across an exposition of umbral calculus in the appendix of a reference. I cannot help but...
2. ### B Problem with the concept of differentiation

We define differentiation as the limit of ##\frac{f(x+h)-f(x)}h## as ##h->0##. We find the instantaneous velocity at some time ##t_0## using differentiation and call it change at ##t_0##. We show tangent on the graph of the function at ##t_0##. But after taking h or time interval as zero to find...
3. ### Calculus Derivative Terminology

I have attached an image of a function that I fit to a scatter plot, and I would like to know if there is a term for the point on the function at which the slope transitions from being less than -1 to greater than -1. I have highlighted this point approximately in yellow...
4. ### Simon suffers an injury at a campsite on the bank of a canal

Homework Statement Simon suffers an injury at a campsite on the bank of a canal that is 6sqrt(2) km wide. if he is able to paddle his kayak at 5km/h and pedal a bicycle at 15km/h along the opposite bank, where should be land on the opposite bank to reach the hospital in minimum time? Homework...
5. ### Correct terminology for taking differentials?

Consider the following two calculations: (1) d(x\cos x)=(\cos x-x\sin x)dx (2) \frac{d}{dx}(x\cos x)=(\cos x-x\sin x) I would describe these both as differentiation. Is there a standard terminology that allows one to make the distinction between the two, if desired? The best I could come up...
6. ### Trying to self-learn calculus but stuck on this idea

So I started with limits and everything went well, and then into derivatives, and a problem started. First I know how to find the derivative of basic things using like ( limit definition of derivative) or (rules for derivatives) or ( product rule/ quotient rule/ chain rule)... My problem...
7. ### Queries regarding Inflection Points in Curve Sketching

Homework Statement Let A be a set of critical points of the function f(x). Let B be a set of roots of the equation f''(x)=0. Let C be a set of points where f''(x) does not exist. It follows that B∪C=D is a set of potential inflection points of f(x). Q 1: Can there exist any inflection points...
8. ### One-forms in differentiable manifolds and differentials in calculus

Suppose that we have this metric and want to find null paths: ds^2=-dt^2+dx^2 We can easily treat dt and dx "like" differentials in calculus and obtain for $$ds=0$$ dx=\pm dt \to x=\pm t Now switch to the more abstract and rigorous one-forms in differentiable manifolds. Here \mathrm{d}t (v)...
9. ### Find Null Paths in Differentiable Manifolds Using One-Forms

Suppose that we have this metric and want to find null paths: ds^2=-dt^2+dx^2 We can easily treat dt and dx "like" differentials in calculus and obtain for $$ds=0$$ dx=\pm dt \to x=\pm t Now switch to the more abstract and rigorous one-forms in differentiable manifolds. Here \mathrm{d}t (v)...
10. ### Finding the derivative of a revenue function

Homework Statement The revenue function for a product is r = 8x where r is in dollars and x is the number of units sold. the demand function is q = -1/4p + 10000 where q units can be sold when selling price is p. what is dr/dp? Homework Equations r=pq The Attempt at a Solution I substituted...
11. ### Derivative as a rate of change exercise

Homework Statement A police car is parked 50 feet away from a wall. The police car siren spins at 30 revolutions per minute. What is the velocity the light moves through the wall when the beam forms angles of: a) α= 30°, b) α=60°, and c) α=70°? This is the diagram...
12. ### How to calculate the derivative in (0, ∞)?

The function f: R → R is: f(x) = (tan x) / (1 + ³√x) ; for x ≥ 0, sin x ; for (-π/2) ≤ x < 0, x + (π/2) ; for x < -π/2 _ For the interval (0,∞), we are interested in f such that f(x) = (tan x) / (1 + ³√x) ; for x ≥ 0 f(x) = tan x / (1 + x¹ʹ³)            (1 + x¹ʹ³)•sec²x −...
13. ### Calculus derivative of given function vanishes at some point between a and b

Question: Show that the derivative of f(x) = (x-a)m (x-b)n vanishes at some point between a and b if m and n are positive integers. My attempt: f(x) = (x-a)m (x-b)n f '(x) = m(x-a)m-1 (x-b)n + n(x-a)m (x-b)n-1 f '(x) = [(x-a)n-1 (x-b)n-1 ] [(m)(x-b) +(n)(x-a)] And this is as far as I got.
14. ### CAlculus derivative help please?

Homework Statement Calculus Derivative help? when x = 0, f=2, f '=1, g=5, g '= -4 when x = 1, f=3, f '=2, g=3, g '= -3 when x = 2, f=5, f '=3, g=1, g '= -2 when x = 3, f=10, f '=4, g=0, g '= -1 based on the above table, if a = f +2g, then a'(3) =? please show steps.thanks Homework...
15. ### Calculus Derivative Question

Homework Statement derivative of (e^x-e^-x)/(e^x+e^-x) Homework Equations The Attempt at a Solution This answer was marked 2/4 on the test, can anyone help me get the right answer please ?
16. ### Calculus Homework: Derivative of g(x) = x * sqrt(4-x) using Product Rule

Homework Statement Find the derivative of g(x)=x * sqrt(4-x)Homework Equations The Attempt at a Solution I know I use the product rule, but I am not sure how to derive the sqrt(4-x) portion.
17. ### Help with Vector Calculus Derivative

Hey... i was hopeing somebody can help me with a homework question... its about vector calc... taking the deriviative. d/dx[r(t) dot r(t)] = r'(t) dot r(t) + r(t) dot r'(t) = 2r'(t) dot r(t) I know it sounds sillly, but i was just wondering how on Earth they got 2r'(t) dot r(t) ?
18. ### Calculus Derivative Question

Hey, Anyone out there able to help me out? I'm trying to find y'' of the equation y=xtanx. I found y' to equal 1(sec²x) but I don't know what to do after that. I know the final answer should be 2cosx + 2xsinx/ cos³x after being simplified and stuff, but I am clueless as to how to get there...