## Derivatives

## What are Derivatives?

**In simple terms, the derivative of a function measures how the output value of a function changes as the input changes.** It is often represented as the slope of the tangent line at any point of the function graph. The derivative at a point tells us the rate at which the function’s value is changing at that point.

## Derivative Formula

**Derivatives** are an essential component of calculus, serving as a powerful tool to measure the sensitivity of one quantity in relation to changes in another. This concept is foundational in analyzing how functions change, which is vital for modeling dynamic systems across various scientific and mathematical disciplines.

The formula for the derivative of a function with respect to a real variable can be expressed mathematically as:

## Derivative Formulas in Calculus

The three foundational categories of derivativesβ**algebraic, logarithmic/exponential, and trigonometric**βare derived from the first principles of differentiation. These categories, which include derivatives of algebraic expressions, logarithms, exponents, and trigonometric functions, form the core of standard derivative formulas, crucial tools in calculus. Each type serves a specific function and is derived by applying the fundamental concept of differentiation. This comprehensive breakdown highlights the essential nature of these derivatives in various mathematical and real-world applications

**Algebraic Functions**: These involve polynomials or rational functions where the derivative is obtained using basic rules such as the power rule.**Logarithmic and Exponential Functions**: The derivatives of functions like ππ₯ and lnβ‘(π₯) are foundational in studying natural growth processes and decay.**Trigonometric Functions**: Derivatives of sine, cosine, and other trigonometric functions are pivotal in physics and engineering for modeling periodic behavior.

### Power Rule of Derivatives

Using the example above, the derivative of π₯Β² is 2π₯. Following this pattern, we can also determine that the derivative of π₯Β³ is 3π₯Β², and the derivative of π₯β΄ is 4π₯Β³ This observation leads us to the **Power Rule** in differentiation, which generalizes this concept for any power of π₯*x*. The Power Rule is formally stated as:

*d/dx*β(*x*βΏ)=*nx*βΏβ»ΒΉ

## Derivatives of Log/Exponential Functions

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

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

The derivative of e^x is, d/dx (eΛ£) = eΛ£

The derivative of a^x is, d/dx (aΛ£) = aΛ£ ln a

## Derivatives of Trigonometric Functions

If y = sin x, y’ = cos x

If y = cos x, y’ = -sin x

If y = tan x, y’ = secΒ² x

If y = cot x, y’ = -cosecΒ² x

If y = sec x, y’ = sec x tan x

If y = cosec x, y’ = -cosec x cot x

## Derivatives of Inverse Trigonometric Functions

Here are the derivatives of trigonometric functions

The derivative of inverse sine is, d/dx (sinβ»ΒΉx) = 1/β(1-xΒ²)

The derivative of inverse cosine is, d/dx (cosβ»ΒΉx) = -1/β(1-xΒ²)

The derivative of inverse tan is, d/dx (tanβ»ΒΉx) = 1/(1 + xΒ²)

The derivative of inverse cot is, d/dx (cotβ»ΒΉx) = -1/(1 + xΒ²)

The derivative of inverse cosec is, d/dx (cscβ»ΒΉx) = -1/ [|x| β(xΒ² – 1) ], x β 1, -1, 0

The derivative of inverse sec is, d/dx (secβ»ΒΉx) = 1/ [|x| β(xΒ² – 1) ], x β 1, -1, 0

## Fundamental Rules of Derivatives

The following are the fundamental rules of derivatives. Let us discuss them in detail.

### Power Rule

The Power Rule simplifies the differentiation of power functions. If **π¦=xβΏ**, then the derivative ππ¦/ππ₯β is given by **ππ₯βΏβ»ΒΉ**. For example, for π(π₯)=π₯β΅, the derivative πβ²(π₯) would be 5π₯β΄

### Sum/Difference Rule

This rule states that the derivative of the sum or difference of two functions is the sum or difference of their derivatives. Mathematically,

**dy/dx [u Β± v]= du/dx Β± dv/dx.**

This rule is particularly useful for handling complex expressions composed of simpler terms.

### Product Rule

The product rule of derivatives states that if a function is a product of two functions, then its derivative is the derivative of the second function multiplied by the first function added to the derivative of the first function multiplied by the second function.

**dy/dx [u Γ v] = u Β· dv/dx + v Β· du/dx. If y = xβ΅eΛ£ ^{ }, we have y’ = x^{5 }. e^{Λ£ }+ e^{Λ£}. 5xβ΄^{ }= e^{Λ£ }(x^{β΅ }+ 5xβ΄)**

### Quotient Rule

When differentiating a quotient of two functions, the Quotient Rule is applied. It is given by **d/dx (u/v) = (v Β· du/dx – u Β· dv/dx)/ v**Β² This formula is essential for functions where one function is divided by another.

### Constant Multiple Rule

This rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. Formally, if π¦=πβ π(π₯), then d/dx [c(f(x)] = c Β· d/dx f(x).Constant Rule

The Constant Rule states that the derivative of a constant value is zero. If π¦=π where *k* is a constant, then ππ¦/ππ₯=0. This result follows logically from the Power Rule as the power of x in a constant is zero.

## Derivatives of Composite Functions (Chain Rule)

If *f* and *g* are differentiable functions within their respective domains, then the composite function π(π(π₯)) is also differentiable. This principle is encapsulated by the chain rule for differentiation, which is essential for handling composite functions. The derivative of the composite function (πβπ)β²(π₯) is given by:

**(πβπ)β²(π₯)=πβ²[π(π₯)]β
πβ²(π₯) **

Alternatively, this can be expressed as: If π¦=π(π’) and π’=π(π₯), then ππ¦/ππ₯=ππ¦/ππ’β ππ’/ππ₯β

For instance, consider the function π¦=tanβ‘Β²(π₯). This function is composite, allowing us to define it in terms of π’, where π’=tanβ‘(π₯)), and thus π¦=π’Β². Calculating the derivatives, we have:

- ππ¦/ππ’=2π’ since the derivative of π’2 with respect to π’ is 2π’.
- ππ’/ππ₯=secβ‘Β²(π₯) because the derivative of tanβ‘(π₯) with respect to π₯ is secβ‘Β²(π₯).

Applying the chain rule, the derivative of π¦ with respect to π₯ is:

ππ¦/ππ₯=ππ¦/ππ’β ππ’/ππ₯=2π’β secβ‘Β²(π₯)=2tanβ‘(π₯)secβ‘Β²(π₯)

## Derivatives of Implicit Functions

In situations where π¦ is a function of π₯ but cannot be explicitly expressed using the variables π₯ and π¦, we use a method called implicit differentiation. This technique is especially useful when the relationship between π₯ and π¦ forms an equation that can’t easily be solved for π¦. To apply implicit differentiation, you start by differentiating both sides of the equation with respect to π₯.

For example, consider the equation 2π₯+π¦=12. To find the derivative of π¦ with respect to π₯, differentiate each term of the equation with respect to π₯:

d/dxβ(2*x*+*y*)=d/dxβ(12)

The derivative of 2*x* with respect to π₯ is 2, and since *y* is a function of π₯, its derivative becomes ππ¦/ππ₯β. The derivative of a constant, like 12, is 0:

2+*dx/dy*β=0

To isolate ππ¦/ππ₯β, solve for it:

ππ¦/ππ₯=β2

## Parametric Derivatives

In a function, we may have the dependent variables x and y which are dependent on the third independent variable. If x = f(t) and y = g(t), then derivative is calculated as dy/dx = f'(x)/g'(x).

## Higher-order Derivatives

Higher-order derivatives refer to the derivatives taken multiple times on a function. The first derivative of a function represents the rate at which the function’s output changes with respect to its input. The second derivative, the derivative of the first derivative, provides information about the curvature or the acceleration of the function. Similarly, third-order derivatives and beyond can also be calculated.

### Notation of Higher-Order Derivatives

The notation for higher-order derivatives can vary. For a function π¦=π(π₯), the first three derivatives are often denoted as:

- First derivative: πβ²(π₯) or ππ¦/ππ₯β
- Second derivative: πβ²β²(π₯) or πΒ²π¦/ππ₯β
- Third derivative: πβ²β²β²(π₯) or πΒ³π¦/ππ₯β

## Partial Derivatives

If u = f(x,y) we can find the partial derivative of with respect to y by keeping x as the constant or we can find the partial derivative with respect to x by keeping y as the constant. Suppose f(x, y) = xΒ³ yΒ² , the partial derivatives of the function are:

βf/βx(xΒ³ yΒ²) = 3xΒ²y and

βf/βy(xΒ³ yΒ²) = xΒ³ 2y

Further, we can find the second-order partial derivatives also like βΒ²f/βxΒ², βΒ²f/βyΒ², βΒ²f/βx βy, and βΒ²f/βy βx.

## Finding Derivative Using Logarithmic Differentiation

In instances where functions are highly complex or involve one function raised to the power of another, such as π¦=π(π₯)α΅β½Λ£βΎ, traditional methods of differentiation may not suffice. To tackle these challenges, we can employ a technique called **logarithmic differentiation**. This method involves taking the natural logarithm (ln) of both sides of the equation, simplifying using logarithmic properties, and then differentiating both sides with respect to π₯. Hereβs how logarithmic differentiation is typically applied:

- Start by taking the natural logarithm of both sides: lnβ‘(π¦)=lnβ‘(π(π₯)π(π₯)).
- Simplify using the logarithm power rule: lnβ‘(π¦)=π(π₯)β lnβ‘(π(π₯))
- Differentiate both sides with respect to π₯, applying the chain rule on the left side and the product rule on the right side.
- Solve for ππ¦/ππ₯β to find the derivative of the original function

## Maxima/Minima by Using Derivatives

Maxima and minima refer to the highest and lowest values, respectively, that a function achieves within a given range. A **maximum** point on a function is where the function value is higher than at any other point nearby, while a **minimum** point is where the function value is lower than at any point in its vicinity. These points can be categorized as either local or global:

**Local maximum (or minimum)**: The function value is the highest (or lowest) in a small surrounding neighborhood.**Global maximum (or minimum)**: The function value is the highest (or lowest) across the entire domain of the function.

### Steps to Find Maxima and Minima Using Derivatives

**Find the derivative**: Calculate the first derivative πβ²(π₯) of the function π(π₯). This derivative represents the rate of change of the function.**Set the derivative to zero**: Solve the equation πβ²(π₯)=0 to find the critical points. These are potential candidates for maxima and minima since the slope of the tangent at these points is zero.**Determine the nature of critical points**: To classify each critical point as a maximum, minimum, or neither, use the second derivative test:- Compute the second derivative πβ²β²(π₯).
- Substitute the critical points into πβ²β²(π₯):
- If πβ²β²(π₯)>0, the function has a local minimum at that point.
- If πβ²β²(π₯)<0, the function has a local maximum at that point.
- If πβ²β²(π₯)=0, the test is inconclusive, and further analysis is required (possibly using higher derivatives or other methods).

## Practice Problems on Derivatives

**Problem 1**: Calculate the derivative of the function: π(π₯)=7π₯β΄

**Solution:** Using the power rule, πβ²(π₯)=4β
7π₯β΄β»ΒΉ=28π₯Β³.

**Problem 2**: Find the derivative of the function: π(π₯)=π₯Β²β
lnβ‘(π₯)

**Solution:**

Using the product rule, πβ²(π₯)=(π₯Β²)β²β lnβ‘(π₯)+π₯Β²β (lnβ‘(π₯))β²

πβ²(π₯)=2π₯β lnβ‘(π₯)+π₯Β²β 1/π₯=2π₯lnβ‘(π₯)+π₯

## FAQs

### What Are Derivatives in Simple Terms?

Derivatives are financial instruments whose value is derived from the performance of underlying assets such as stocks, bonds, commodities, or market indexes. They are used primarily for hedging risk or speculating on price movements of the underlying assets.

### What Are the 4 Main Derivatives?

The four main types of derivatives are:

**Futures**: Contracts to buy or sell an asset at a predetermined future date and price.**Options**: Contracts that give the right, but not the obligation, to buy or sell an asset at a set price before a certain date.**Swaps**: Agreements to exchange one set of cash flows for another.**Forwards**: Customized contracts to buy or sell an asset at a set price and date, typically used by institutions.

### What Is the Basic Concept of Derivatives?

The basic concept of derivatives is to provide a way to manage financial risk by allowing investors to hedge against potential losses or speculate on changes in the underlying asset’s price. Derivatives set terms for buying or selling assets at future dates, based on predictions of asset price movements.

### What Is Derivative Used in Real Life?

In real life, derivatives are used in various ways:

**Businesses**use them to hedge against changes in commodity prices or currency exchange rates.**Investors**use them to speculate on price movements without actually owning the underlying asset.**Insurance companies**use them to manage risk exposure.

### How Do You Make Money from Derivatives?

Making money from derivatives involves strategies like hedging, which protects against losses in other investments, and speculation, where traders predict market movements to profit from price discrepancies. Success in derivatives trading requires understanding market conditions and the behavior of the underlying assets.