Derivatives

In calculus, a derivative represents how a quantity changes with respect to another. Derivatives measure the rate at which one variable, say, y changes as another variable, x, changes. This concept is fundamental in understanding the behaviour of a function and is widely used across various fields.

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What is a Derivative?

The derivative of a function provides insight into how the function’s output changes as its input changes. Mathematically, it is the differential coefficient of y with respect to x. Also, the derivative tells us the slope of the function at any given point. It indicates if the function is increasing or decreasing and at what rate.

Differentiation

Differentiation is the process used to find the derivative of a function. This involves applying various rules and techniques to calculate how a function’s output varies with changes in its input.

How to Find a Derivative

To find a derivative, we follow specific rules and methods. Below given are some basic rules:

• Power rule: if f(x) = xn then, f'(x) = n xn-1

• Product rule: if f(x) = u(x) . v(x) then, f'(x) =u'(x) . v(x) + u(x) . v'(x)

• Quotient rule: if f(x) = u(x)v(x) then, f'(x) = u'(x) . v(x) – u(x) . v'(x){v(x)}2

• Chain rule: if f(x) = g(h(x)) then, f'(x) = g'(h(x)) . h'(x)

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Representation of the Derivative

The derivative of a function is often denoted in several ways:

• ddx f(x)
• dfdx
• Df(x)
• f'(x)

Each notation represents the same concept but is used in different contexts or preferences.

Therefore, the derivative of a function captures the instantaneous rate of change of the function with respect to x. It is found by considering the slope of the tangent line as the interval between two points on the function becomes infinitesimally small. This concept is fundamental in calculus and is widely used in various applications to analySe and interpret the behaviour of functions.

Limit of the Derivative

The formula for finding the derivative,

f'(x) = h0f(x+h)−f(x)h

is commonly referred to as the “limit definition of the derivative” or “derivative by the first principle.” This fundamental concept in calculus provides a precise way to determine the rate of change of a function at a specific point.

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Applications and Importance of Limits

The limit definition of the derivative is not just a theoretical concept but is widely used in various applications of calculus, including physics, engineering, economics, and more. It provides the foundation for more advanced differentiation techniques and helps in analysing the behaviour of the functions, optimising processes, and solving real-world problems.

Interpretation of Derivatives

In mathematics, the derivative of a function f(x), denoted as f′(x), provides valuable insights into the function’s behaviour at specific points. Here’s how derivatives can be interpreted in various contexts:

Slope of the Tangent Line:

The derivative at a particular point on the function represents the slope of the tangent line to the curve at that point. This slope indicates how steep the curve is and the direction it is heading at that specific location.

Instantaneous Rate of Change:

The derivative gives the instantaneous rate of change of the function with respect to its variable. This is crucial for understanding how the function’s value changes at any given moment, rather than just over an interval.

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The velocity of a Particle:

In physics, if f(x) represents the displacement of a particle over time, the derivative f′(x) represents the particle’s velocity. This tells us how quickly the particle’s position is changing at any given time.

Optimisation:

Derivatives are used to find the maximum and minimum values of a function. By setting the derivative equal to zero and solving for the variable, we can identify critical points where the function might reach its highest or lowest values.

Function Behavior:

Derivatives help in determining intervals where the function is increasing or decreasing. If the derivative is positive, the function is increasing and if it is negative, then the function is decreasing.

Derivative Using the First Principle

To find the derivative of a function using the first principle (limit definition), we use the following formula: f'(x) = h0f(x+h)−f(x)h

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Basic Derivative Formulas

If f(x) = xn then, f'(x) = n xn-1

If f(x) = u(x) . v(x) then, f'(x) =u'(x) . v(x) + u(x) . v'(x)

If f(x) = u(x)v(x) then, f'(x) = u'(x) . v(x) – u(x) . v'(x){v(x)}2

If f(x) = g(h(x)) then, f'(x) = g'(h(x)) . h'(x)

If y = sin x, y’ = cos x

If y = cos x, y’ = -sin x

If y = tan x, y’ = sec2x

If y = cot x, y’ = -cosec2 x

If y = sec x, y’ = sec x tan x

If y = cosec x, y’ = -cosec x cot x

The derivative of ln x is, d/dx (ln x) = 1/x

The derivative of log x is, d/dx (loga x) = 1/(x ln a)

The derivative of ex is, d/dx (ex) = ex

The derivative of ax is, d/dx (ax) = ax ln a

The derivative of inverse sine is, d/dx (sin-1x) = 1/√(1- x2)

The derivative of inverse cosine is, d/dx (cos-1x) = – 1/√(1- x2)

The derivative of inverse tan is, d/dx (tan-1x) = 1/(1+ x2)

The derivative of the inverse cot is, d/dx (cot-1x) = -1/(1+ x2)

The derivative of inverse cosec is, d/dx (cosec-1x) = -1/[|x| √( x2 – 1) ], x ≠ 1, -1, 0

The derivative of inverse sec is, d/dx (sec-1x) = 1/[|x| √( x2 – 1) ], x ≠ 1, -1, 0

Important Notes on Derivatives

• A derivative represents the rate of change of one quantity with respect to another.
• The derivative of a function is defined as f'(x) = h0f(x+h)−f(x)h

Derivatives: FAQs

What is a derivative in calculus?

A derivative in calculus measures how a function changes as its input changes. It represents the rate of change or slope of the function at any given point. For a function f(x), its derivative f′(x) gives the slope of the tangent line to the curve of f(x) at that point.

How do you find the derivative of a function?

To find the derivative of a function, you can use various rules depending on the function type. For example, use the power rule for functions of the form xn, where the derivative is n.xn-1. For more complex functions, you might use the product rule, quotient rule, or chain rule. The limit definition of a derivative can also be used for a rigorous approach.