What's the relationship between a function and its derivative?

Answer 1

the derivative show the rate in which the function value changes by the changes of the function variable

for example
f(x)=2*x
df/dx=2

f(0)=0
f(1)=2
f(2)=4
for each increase in x by 1 the function output increase by 2 which is the derivative

Answer 2

The derivative of a function is the slope, or gradient, of the tangent at any given point on the function, provided that a tangent can be drawn. If not, then the derivative does not exist at that point.

Answer 3

the derivative is a measure for how much the function is raising.

for instance f(x) = x. This is just a line with a slope of 1
the derivative f' = 1

Answer 4

If f is a function, then the mean value of f over a given interval is equal to the limit of the average sum of its derivative f' taken over the interval with n partitions as n approaches infinity.

Answer 5

Let's say we have the function f(x) and its derivative f'(x). Now, if we draw the graph of y = f(x), and draw a tangent line to f(x) at the line (a,f(a)), then the slope of that tangent line is f'(a).

In simpler terms, the derivative of a function is the equation of the slope of the tangent lines within its domain.

^_^

Answer 6

in one variable, where f'(x) is the derivative of f,

f'(x) = lim h->0 { [f(x+h) - f(x)] / h }

where h is a small change, i.e. delta*x in many notations. It can be though of as the rate of change of the function at some point, i.e. the gradient to the tangent between 'close' points

Answer 7

function= integral of its derivative

Answer 8

the first by-product is the speed of replace of the function. operating example, if the x-axis (horizontal axis) represents time, and the y-axis(vertical axis) represents distance traveled, then the first by-product represents the quantity of distance traveled in an infinitely small time, i.e. the speed. the 2d by-product would represent the speed of replace of speed in line with time, i.e. acceleration. The graph of the first by-product may be the graph of the speed vs time, and the 2d by-product may be the graph of acceleration vs time. it really is nice even as first gaining comprehend-how of those thoughts to narrate them to actual examples each and every time accessible. in spite of the indisputable fact that, at an extremely stepped ahead aspect it really is not accessible, in spite of the indisputable fact that it really is way into your destiny. besides, good luck interpreting Calculus. i'm hoping that you savor it as a lot as I did because it really is interesting.