In the field of mathematics and even more specifically calculus, derivatives are one of the most fundamental concepts used to measure the rates of change within a system.
Derivatives explain how a function’s output will change depending on its input, this forms the foundation for many mathematical and scientific applications, from understanding motion, all the way to modeling very complex systems in economics, physics and biology.
In this article, we shall walk you through the concept of derivatives, giving you and in-depth understanding of all the different rules of differentiation with examples and a step by step process to a solution to help practically inform you. Let’s dive in!
What is a Derivative?
So, what exactly is a derivative?
Simply put, a derivative is a representation of the rate at which a function changes. For example, if f(x) represents the position of a car over time, f'(x) (also said as the derivative of x) represents the car’s velocity.
Mathematically, the derivative is defined as:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) – f(x)}{h} \]
This limit finds the slope of the tangent line at any point on a curve, helping us analyze how quickly f(x) changes near that point.
Now that we’ve understood what derivatives are, let’s get to the rulings surrounding it.
Rules of Differentiation
Derivatives all follow a list of rules. These rules govern how a function is differentiated depending on its own unique operations.
The Power Rule
The power rule simplifies differentiating functions of the form x^n. The power rule states:
\[ \frac{d}{dx}[x^n] = n \cdot x^{n-1} \]
Example:
Differentiate \[ f'(x) = 3x^{3-1} = 3x^2 \].
The Constant Rule and Constant Multiple Rule
The constant rule states that the derivative of a constant is always zero. This is given by:
\[ \frac{d}{dx}[c] = 0 \]
If a constant multiplies a function, differentiate the function and multiply by the constant:
\[ \frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}[f(x)] \].
Example:
\[ \frac{d}{dx}[7x^2] = 7 \cdot 2x = 14x \].
The Sum and Difference Rules
To differentiate sums or differences of functions:
\[ \frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}[f(x)] \pm \frac{d}{dx}[g(x)] \].
Example:
\[ \frac{d}{dx}[x^2 + 3x] = 2x + 3 \].
The Product Rule of differentiation
When multiplying two functions, the product rule applies:
\[ \frac{d}{dx}[f(x) \cdot g(x)] = f(x) \cdot g'(x) + f'(x) \cdot g(x) \].
Example:
\[ \frac{d}{dx}[(x^2)(e^x)] = x^2 \cdot e^x + 2x \cdot e^x = e^x(x^2 + 2x) \].
The Quotient Rule
When dividing, use the quotient rule given below:
\[ \frac{d}{dx} \left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) – f(x)g'(x)}{[g(x)]^2} \].
Example:
\[ \frac{d}{dx} \left[\frac{x^2 + 1}{x}\right] = \frac{2x \cdot x – (x^2 + 1) \cdot 1}{x^2} = \frac{x^2 – 1}{x^2} \].
The Chain Rule
For composite functions, use the chain rule:
\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \].
Example:
\[ \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x = 2x \cos(x^2) \].
Higher-Order Derivatives
Higher-order derivatives measure repeated rates of change. For example, the second derivative \(f”(x)\) of a position function represents acceleration.
Example:
Find the second derivative of \[ f'(x) = 3x^2, \quad f”(x) = 6x \].
Applications of Derivatives
Finding Tangents and Normals
To find the slope of the tangent to a curve, differentiate the function and evaluate it at the point of interest.
Example:
Find the tangent to \[ y – y_1 = m(x – x_1) \].
\[ f'(x) = 2x + 2, \quad f'(1) = 4 \].
At x = 1, the slope of the tangent line is 4. The point on the curve is (1, 3). The equation of the tangent line is:
\[y – y_1 = m(x – x_1)\].
Substituting:
\[ y – 3 = 4(x – 1), \quad y = 4x – 1 \].
Optimization Problems
Derivatives are vital in finding the maximum or minimum values of functions. The process involves differentiating the function, setting the derivative equal to zero, and analyzing the second derivative to determine the nature of these points.
Example:
Find the maximum area of a rectangle under the curve \(y = 6 – x^2\) in the first quadrant.
\[ A(x) = x \cdot y = x(6 – x^2) = 6x – x^3 \].
Differentiate to find critical points:
\[ A'(x) = 6 – 3x^2, \quad A”(x) = -6x \].
\[ A(\sqrt{2}) = 6\sqrt{2} – (\sqrt{2})^3 = 4\sqrt{2} \].
Use the second derivative to confirm:
\(A”(x) = -6x\).
At \(x = sqrt{2}, A”(x) < 0\), indicating a maximum.
The maximum area is \(A(sqrt{2}) = 4sqrt{2}\).
In conclusion
Derivatives give a systematic way to analyze rates of change and make the best of functions. From the simple of rules of differentiation to more advanced applications, getting to understanding derivatives will give you fundamental knowledge on your way to solving real-world problems in physics, engineering, and beyond.
Now that we’ve studied derivation, let’s tease you on the next essential topic on your journey—integrals.
In calculus, derivatives and integrals are two sides of the same coin. While derivatives measure change, integration calculates the accumulation of quantities. In the upcoming article, we’ll explore integration, its rules, and its connection to derivatives through the fundamental theorem of calculus. Stay tuned!
