Derivative – First Order

The first order derivative of a function represents the rate of change of one variable with respect to another variable. For example, in Physics we define the velocity of a body as the rate of change of the location of the body with respect to time.  Here location is the dependent variable on the other hand time is the independent variable.  To find the velocity, we need to compute the first order derivative of the location. Similarly, we can this concept for computing rate of dependency of one variable over the other. Let us discuss Derivative – First Order in detail.

Derivative – First Order

It’s another name is first order differentiation. Differentiation is the algebraic method of finding the derivative for a function at any given point. The derivative is a root concept of calculus.

There are two ways of introducing this concept, first one is the geometrical way, and another one is the physical way. The slope of a curve translates to the rate of change when looking at practical examples in real life.

Thus both the slope and the instantaneous rate of change are equivalent here. And derivative is the method to find both of these at any point.

The first order derivative mainly tells us that in which direction the function is going. That is whether the function is increasing or decreasing with respect to some given reference.

The first order derivative also represents the instantaneous rate of change of some dependent variable with respect to an independent variable. In terms of the graph, it gives the slope of the tangent line drawn at a given point on the curve of the graph.

When we take the derivative of a function, we end up with another function that gives the slope of the original function. The derivative of a function should have the same limit from left to right for it to be differentiable at that point.

The derivative also tells us the rate of change from one quantity compared to another. If we know how much distance a car has traveled over time, the derivative can tell us it’s velocity and hence the rate of change of velocity will give acceleration at any point in time.

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Slope of the Secant Line

The slope of the secant line is given as:

m_PQ.secant = \(\frac{f(a+h) – f(a)}{(a+h) – a}\)

= \(\frac{f(a+h) – f(a)}{h}\)

As Q approaches closer to P, the limiting portion is called the tangent line. In terms of the limit theory of calculus, the slope of the tangent line m_PQ.tangent will be the limiting value of m_PQ.secant as h → 0.

i.e.

m_PQ.tangent = lim _{h → 0} \(\frac{f(a+h) – f(a)}{h}\)

Fundamental Definition of First-order Derivative of a Function

The first order derivative of a function f(x) at x=a is defined as:

f ‘(a)= lim _{h → 0} \(\frac{f(a+h) – f(a)}{h}\)

Another definition is:

The derivative of the function f(x)  at x=a  is defined as:

f ‘(a)= lim _x→a \(\frac{f(x) – f(a)}{x-a}\)

As x→a , it is nothing but the slope of the tangent line at P.

Other Notations of Derivatives

Derivate can be denoted in several ways.  For the first derivative, the notations are

f ‘(x), \(\frac{d}{dx}\) f(x), y ‘, and \(\frac{dy}{dx}\)

Source: freepik.com

Differentiable and Non Differentiable

We must be careful when finding the derivative because it is not possible to differentiate every function. Most functions are differentiable, i.e. a derivative exists at every point on the curve of the function. Some functions, however, are not completely differentiable.

Some Theorems of Differentiation

Theorem-1:

The derivative of a constant is zero.  If f(x) = k , where k  is a constant, f ‘(x) = 0.

Theorem-2:

The derivative of f (x) = xⁿ where n≠ 0 is f ‘(x) = nx^n-1 .

Theorem-3:

The derivative of f(x) = kg(x), where k is a constant is f ‘ (x)= kg'(x).

Theorem-4:

The derivative of f(x) = u(x)±v(x) is f ‘(x) = u'(x)  ± v'(x)

Theorem-5:

The derivative of  f(x) = u(x)v(x) is f ‘(x) = u(x) \(\frac{d}{dx}\) v(x) + v(x) \(\frac{d}{dx}\) u(x).

This is Product Rule.

Theorem-6:

The derivative of

f(x) = \(\frac{u(x)}{v(x)}\) is f ‘(x) = \(\frac{v(x) \(\frac{d}{dx}\) u(x) – u(x) \(\frac{d}{dx}\) v(x)}{[v(x)]²}

This is Quotient Rule.

Solved Question on Derivative – First Order

Find the slope of the tangent line of the curve y = 4x² at point (3,36).

m = lim _{h → 0} \(\frac{f(3+h) – f(3)}{h}\)

= lim _{h → 0} \(\frac{4(3+h)² – 4(3)²}{h}\)

= lim _{h → 0} \(\frac{4(9+h² + 6h) – 36}{h}\)

= lim _{h → 0} \(\frac{36 +4h² + 24h – 36}{h}\)

= lim _{h → 0} \(\frac{h(4h+ 24}{h}\)

= lim _{h → 0} 4h + 24

=24