Derivatives

Derivatives form an important quality of calculus, capturing the essence of change and motion. Whether you are exploring the slopes of curves or the rates at which quantities change, derivatives offer powerful tools for analysis. This article delves into the fundamentals of derivatives, explores various types, and dives into some advanced concepts like implicit differentiation and the derivatives of trigonometric functions.

1.0What are Derivatives?

At its core, a derivative represents the rate of change of a function with respect to a variable. If we consider a curve on a graph, the derivative at any given point gives us the slope of the tangent line at that point. Mathematically, if y = f(x), then the derivative of y with respect to x is denoted as $\frac{dy}{dx}or{f}^{\prime }(x).$

2.0Types of Derivatives

Derivatives can be classified based on their nature and the functions they are derived from. Here are the four primary types:

1. Ordinary Derivatives: These are derivatives of functions with respect to a single variable. For example, $\frac{dy}{dx}$ is the ordinary derivative of y with respect to x.
2. Partial Derivatives: When dealing with functions of multiple variables, such as f (x, y, z), we use partial derivatives. Here, each derivative represents the rate of change with respect to one variable while keeping the others constant, denoted as $\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}$, etc.
3. Higher-Order Derivatives: These are derivatives taken multiple times. The second derivative, $\frac{d^{2}y}{d{x}^2}$ , represents the rate of change of the first derivative and is particularly useful in studying concavity and acceleration.
4. Implicit Derivatives: Sometimes, functions are given implicitly rather than explicitly. For example, if x² + y² = 1, finding $\frac{dy}{dx}$ involves implicit differentiation.

3.0Derivatives Basic Rules

To compute derivatives, certain fundamental rules must be followed:

• Power Rule: If f(x) = xⁿ , then f '(x) = nx^n–1.
• Product Rule: If u(x) and v(x) are functions of x, then

$\frac{d}{dx}[u(x)v(x)]=u^{\prime }(x)v(x)+u(x)v^{\prime }(x)$

• Quotient Rule: For two functions u(x) and v(x), 

$\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right]=\frac{u^{\prime }(x)v(x)-u(x)v^{\prime }(x)}{v(x{)}^2}$

• Chain Rule: If a function y is composed of another function u such that y = f(u(x)), then $\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}$.

4.0Derivatives of Trigonometric Functions

Trigonometric functions are spread widely in calculus, and their derivatives follow specific patterns:

• $\frac{d}{dx}(\sin x)=\cos x$
• $\frac{d}{dx}(\cos x)=-\sin x$
• $\frac{d}{dx}(\tan x)={\sec }^{2}x$
• $\frac{d}{dx}(\mathrm{cosec}x)=-\mathrm{cosec}x\cot x$
• $\frac{d}{dx}(\sec x)=\sec x\tan x$
• $\frac{d}{dx}(\cot x)=-{\mathrm{cosec}}^{2}x$

5.0Derivatives of Inverse Trigonometric Functions

Inverse trigonometric functions also have well-defined derivatives:

• $\frac{d}{dx}(\arcsin x)=\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}(\arccos x)=-\frac{1}{\sqrt{1-x^2}}$
• $\frac{}{dx}(\arctan x)=\frac{1}{1+x^2}$
• $\frac{d}{dx}(\mathrm{arccosec}x)=-\frac{1}{|x|\sqrt{x^2-1}}$
• $\frac{d}{dx}(\mathrm{arcsec}x)=\frac{1}{|x|\sqrt{x^2-1}}$
• $\frac{d}{dx}(\mathrm{arccot}x)=-\frac{1}{1+x^2}$

These derivatives are essential in various applications, including integrals and solving differential equations.

6.0Second Derivative and Implicit Differentiation

The second derivative, or the derivative of the derivative, provides insights into the concavity and inflection points of a function. For example, if y = f(x), then $\frac{d^{2}y}{d{x}^2}$ indicates whether the curve is concave up or down at a given point.

Implicit differentiation is particularly useful when dealing with equations where the dependent and independent variables are not explicitly separated. For example, given an equation like x² + y² = r², differentiating both sides with respect to x gives us the derivative of y implicitly. This technique is essential for solving problems involving curves that are not easily expressed as functions.

7.0Solved Example of Derivatives

Example 1: Differentiate the function f(x) = 3x⁴ – 5x³ + 2x² – 7x + 10.

Solution:

Using the power rule $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$ :

$f^{\prime }(x)=\frac{d}{dx}\left(3x^4\right)-\frac{d}{dx}\left(5x^3\right)+\frac{d}{dx}\left(2x^2\right)-\frac{d}{dx}(7x)+\frac{d}{dx}(10)$

$f^{\prime }(x)=12x^3-15x^2+4x-7$

So, the derivative of f(x) is f '(x) = 12x³ – 15x² + 4x – 7.

Example 2: Differentiate g(x) = sin x + cos x.

Solution:

Using the derivatives of trigonometric functions:

g^{\prime}(x)=\frac{d}{d x}(\sin x)+\frac{d}{d x}(\cos x)

g'(x) = cos x – sin x

So, the derivative of g(x) is g'(x) = cos x – sin x.

Example 3: Differentiate h(x) = arcsin x.

Solution:

Using the derivative of the arcsine function:

$h^{\prime }(x)=\frac{d}{dx}(\arcsin x)=\frac{1}{\sqrt{1-x^2}}$

So, the derivative of h(x) is $h^{\prime }(x)=\frac{1}{\sqrt{1-x^2}}$.

Example 4: Given the equation x² + y² = 25, find $\frac{dy}{dx}$ using implicit differentiation.

Solution:

Differentiate both sides with respect to x:

$\frac{d}{dx}\left(x^2+y^2\right)=\frac{d}{dx}(25)$

Using the chain rule:

2 x+2 y$\frac{dy}{dx}=0$

Solving for $\frac{dy}{dx}$ :

$2y\frac{dy}{dy}=-2x$

$\frac{dy}{dx}=-\frac{x}{y}$

So, $\frac{dy}{dx}=-\frac{x}{y}$ .

Example 5: For the function f(x) = x³ – 3x² + 2x, find the second derivative $\frac{d^{2}y}{d{x}^2}$.

Solution:

First, find the first derivative:

$f^{\prime }(x)=\frac{d}{dx}\left(x^3-3x^2+2x\right)=3x^2-6x+2$

Now, differentiate f'(x) to find the second derivative:

$f^{\prime \prime }(x)=\frac{d}{dx}\left(3x^2-6x+2\right)=6x-6$

So, the second derivative is f''(x) = 6x – 6.

8.0Practice Question on Derivatives

Differentiate the following functions with respect to x:

• $f(x)=5x^3-4x^2+7x-9$
• $g(x)=\frac{3}{x}+2x^2-\sqrt{x}$
• $h(x)=e^{2x}+\ln (x)$
• $y=sin(3x)$
• z = cos²(x) + sin²(x) 
• x² + y² = 16
• y = x⁴ – 4x³ + 2x²
• f(x) = (3x² + 2x) (x³ – 1)
• e^y + xy = x²
• sin (x + y) = x² + y²

9.0Sample Question on Derivatives

1. What are higher-order derivatives?

Ans: Higher-order derivatives are derivatives of a derivative. For example, the second derivative $\frac{d^{2}y}{d{x}^2}$ is the derivative of the first derivative $\frac{dy}{dx}$. Higher-order derivatives can provide information about the concavity and inflection points of a function.