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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 ...
- Secant Lines
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- Product Rule
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- Differentiate Products
Find the derivatives of products of basic functions.
- Secant Lines
- What Is Differentiation in Maths
- Differentiation Formulas
- Differentiation Rules
- Real-Life Applications of Differentiation
- Solved Examples
- Video Lesson on Class 12 Important Calculus Questions
In Mathematics, Differentiation can be defined as a derivative of a function with respect to an independent variable. Differentiation, in calculus, can be applied to measure the function per unit change in the independent variable. Let y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by: dy / dx If the fu...
The important Differentiation formulasare given below in the table. Here, let us consider f(x) as a function and f'(x) is the derivative of the function. Also, see:
The basic differentiation rules that need to be followed are as follows: 1. Sum and Difference Rule 2. Product Rule 3. Quotient Rule 4. Chain Rule Let us discuss all these rules here.
With the help of differentiation, we are able to find the rate of change of one quantity with respect to another. Some of the examples are: 1. Acceleration: Rate of change of velocity with respect to time 2. To calculate the highest and lowest point of the curve in a graph or to know its turning point, the derivative function is used 3. To find tan...
Q.1: Differentiate f(x) = 6x3 – 9x + 4 with respect to x. Solution: Given: f(x) = 6x3 – 9x + 4 On differentiating both the sides w.r.t x, we get; f'(x) = (3)(6)x2– 9 f'(x) = 18x2– 9 This is the final answer. Q.2: Differentiate y = x(3x2– 9) Solution: Given, y = x(3x2– 9) y = 3x3– 9x On differentiating both the sides we get, dy/dx = 9x2– 9 This is t...
Practice Problems
1. Find the derivative of the function f(x) = 3 sin x + cos x – tan x. 2. Perform the differentiation for the following functions: (i) f(x) = x3 sin 2x (ii) g(x) = 4xe2x− 9x 3. Find the derivative of the function f(x) = x/(x – 2) (i) Using the limit definition of differentiation (ii) Using the quotient rule To know more about Differentiation and any Maths related topics, please visit us at BYJU’S.
It means that, for the function x 2, the slope or "rate of change" at any point is 2x. So when x=2 the slope is 2x = 4, as shown here: Or when x=5 the slope is 2x = 10, and so on. Note: f’ (x) can also be used to mean "the derivative of": f’ (x) = 2x. "The derivative of f (x) equals 2x". or simply "f-dash of x equals 2x".
Aug 17, 2024 · derivative the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative difference quotient. of a function \(f(x)\) at \(a\) is given by \(\dfrac{f(a+h)−f(a)}{h}\) or \(\dfrac{f(x)−f(a)}{x−a}\) differentiation the process of taking a derivative instantaneous rate of ...
Nov 20, 2021 · 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.
Nov 16, 2022 · Section 3.1 : The Definition of the Derivative. In the first section of the Limits chapter we saw that the computation of the slope of a tangent line, the instantaneous rate of change of a function, and the instantaneous velocity of an object at x = a x = a all required us to compute the following limit. lim x→a f (x) −f (a) x −a lim x ...
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Oct 26, 2024 · John L. Berggren. Differentiation, in mathematics, process of finding the derivative, or rate of change, of a function. Differentiation can be carried out by purely algebraic manipulations, using three basic derivatives, four rules of operation, and a knowledge of how to manipulate functions.
