MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping
Menu
shahrukh sufiyaan Grade: 11
        

what is dy/dx? how to understand it correctly and easily?

7 years ago

Answers : (1)

Askiitian.Expert Rajat
24 Points
										

Hi,


In calculus, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position (or distance) of a vehicle with respect to time is the instantaneous velocity (respectively, instantaneous speed) at which the vehicle is traveling.

For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point.

The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration.


Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x. In more precise language, the dependence of y upon x means that y is a function of x. If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point. This functional relationship is often denoted y = ƒ(x), where ƒ denotes the function.


The simplest case is when y is a linear of x, meaning that the graph of y against x is a straight line. In this case, y = ƒ(x) = m x + c, for real numbers m and c, and the slope m is given by


m={\mbox{change in } y \over \mbox{change in } x} = {\Delta y \over{\Delta x}}

where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in." This formula is true because


y + Δy = ƒ(x+ Δx) = m (x + Δx) + c = m x + c + m Δx = y + mΔx.

It follows that Δy = m Δx.


This gives an exact value for the slope of a straight line. If the function ƒ is not linear (i.e. its graph is not a straight line), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.


Rate of change as a limiting value



The tangent line at (x, ƒ(x))





such an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written
 \frac{dy}{dx} \,\!

suggesting the ratio of two infinitesimal quantities.


The secant to curve y= ƒ(x) determined by points (x, ƒ(x)) and (x+h, ƒ(x+h))



The tangent line as limit of secants



the difference between the x values, that is,
m = \frac{\Delta f(x)}{\Delta x} = \frac{f(a+h)-f(a)}{h}.

The derivative is the value of the difference quotient as the secant lines approach the tangent line. Formally, the derivative of the function ƒ at a is the limit


f'(a)=\lim_{h\to 0}{f(a+h)-f(a)\over h}

of the difference quotient as h approaches zero, if this limit exists. If the limit exists, then ƒ is differentiable at a. Here f′ (a) is one of several common notations for the derivative 


 


Regards,


Rajat


Askiitian Expert





7 years ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies
  • Complete JEE Main/Advanced Course and Test Series
  • OFFERED PRICE: Rs. 15,900
  • View Details
  • Statistics and Probability
  • OFFERED PRICE: Rs. 636
  • View Details

Ask Experts

Have any Question? Ask Experts

Post Question

 
 
Answer ‘n’ Earn
Attractive Gift
Vouchers
To Win!!! Click Here for details