Derivative of a function at a point

In very basic terms, the derivative of a function f(x) is the rate of change of the function and is indicated by either f'(x) or df/dx.

Δx denotes a change in the value of x. We continue to shrink the gap between x and (x + Δx) until it is minuscule. As a result, we have the limit Δx—->0. The numerator f(x + Δx) – f(x) denotes the change in the value of the function f throughout the interval Δx. This determines the rate of change of a function f at a location x.

A critical element to remember is that x, and the change in x, may be either positive or negative. As a result, we have 0<|Δx|<e, which is an infinitesimally tiny value.

Finding slopes with derivatives

We are all familiar with the procedure for determining the slope of a straight line. Divide the change in y by the change in x, or, as our primary school maths teacher put it, “rise over run.” This is often referred to as the pace of change. While this method is useful for finding the slope of straight lines, what happens when we need to estimate the slope of a nonlinear function? Will the same method be successful? No, it will not. The derivative, which is the instantaneous rate of change or slope of a function at a certain point, might assist us in overcoming this obstacle.

Formula and definition

Linear equations have consistent slopes over the whole line. Intuitively, you may reason that, since we are examining a curve, the slope of the function will not be constant throughout. That is a reasonable assumption.

When trying to determine the slope of a curve, we are looking for an equation that yields the slope of a line tangent to the curve at any given value of x. This is referred to as the function’s derivative. As mentioned earlier, the derivative is the instantaneous rate of change or slope of a function at a certain position. It returns the precise slope at a given location along the curve.

The derivative is symbolised by the symbol (dy/dx), which stands for the derivative of y in relation to x. Recall that the following formula is used to get the derivative:

dy/dx = limh->0 f(x + h) – f(x)/h

The Derivative of a Function at a Point

It is defined as a function’s instantaneous rate of change at a location expressed in terms of its average rate of change across relevant intervals. Additionally, we refer to “the instantaneous rate of change of f at a as “the derivative of f at a,” with this value denoted by the shorthand notation f′(a). 

We provide the following definition in detail.

Assume that f is a function and that x = a is a value in the domain of the function. The derivative of f with respect to x evaluated at x = a, denoted f′(a), is defined by the formula f’(a) = limh->0 f(a + h) – f(a)/h given that this constraint exists. We read the symbol f′(a) aloud as “f – prime at a” or “f’s derivative assessed at x = a.” 

The following chapters will elaborate on discussing, calculating, applying, and interpreting derivatives. For the time being, we make the following critical points.

The derivative of f at x = a is defined as the limit of f’s average rate of change on the interval [a, a + h] expressed in terms of h–>0. Because this limit is not guaranteed to exist, not every function has a derivative at every point.

A function with a derivative at x = a is differentiable at x = a. The derivative generalises the instantaneous velocity of a position function: while y = s(t) represents the location of a moving body, s′(a) indicates the body’s instantaneous velocity at time t = a.

Conclusion

Calculus is founded on the concept of instantaneous rate of change of a function. It is an extension of the concept of instantaneous velocity that quantifies the rate of change of a certain function at a given place. The slope of the tangent line to the graph of a function at a point is referred to as its derivative.