101 LearnersLast updated on July 16th, 2025
Derivative of f(x)
We use the derivative of f(x) as a tool to understand how the function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now talk about the derivative of f(x) in detail.
What is the Derivative of f(x)?
We now understand the derivative of f(x). It is commonly represented as d/dx (f(x)) or (f(x))'. The function f(x) has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Function Definition: The expression f(x) defines the function.
Differentiability: The function f(x) is differentiable if its derivative exists at all points in its domain.
Derivative Notation: The derivative is represented as f'(x) or d/dx (f(x)).
Derivative of f(x) Formula
The derivative of f(x) can be denoted as d/dx (f(x)) or (f(x))'. The formula we use to differentiate f(x) depends on the specific form of the function.
For example, if f(x) is a polynomial, we use standard differentiation rules. The formula applies to all x where the function is defined and differentiable.
Proofs of the Derivative of f(x)
We can derive the derivative of f(x) using proofs. To show this, we will use the rules of differentiation and mathematical principles.
There are several methods we use to prove this, such as:
- By First Principle
- Using Chain Rule
- Using Product Rule
We will now demonstrate that the differentiation of f(x) results in its derivative using the above-mentioned methods:
- By First Principle The derivative of f(x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of f(x) using the first principle, we will consider f(x). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h
- Using Chain Rule To prove the differentiation of f(x) using the chain rule, we consider the composition of functions if applicable and apply the chain rule appropriately.
- Using Product Rule We use the product rule when f(x) can be expressed as a product of two functions. The product rule formula is: d/dx [u.v] = u'.v + u.v'
Higher-Order Derivatives of f(x)
When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.
To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like f(x).
For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x) is the result of the second derivative and this pattern continues.
For the nth Derivative of f(x), we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change.
Special Cases:
There may be points where the derivative is undefined due to discontinuities or vertical asymptotes. When x is in a domain where f(x) is defined and differentiable, the derivative can be calculated as specified by the differentiation rules.
Common Mistakes and How to Avoid Them in Derivatives of f(x)
Students frequently make mistakes when differentiating f(x). These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:
Mistake 1
Not simplifying the equation
Students may forget to simplify the equation, which can lead to incomplete or incorrect results. They often skip steps and directly arrive at the result, especially when solving using the product or chain rule. Ensure that each step is written in order. Students might think it is awkward, but it is important to avoid errors in the process.
Mistake 2
Forgetting the Undefined Points of f(x)
They might not remember that f(x) can be undefined at certain points. Keep in mind that you should consider the domain of the function that you differentiate. It will help you understand that the function is not continuous at certain points.
Mistake 3
Incorrect use of Differentiation Rules
While differentiating functions, students may misapply differentiation rules. For example: Incorrect differentiation can occur by not properly applying rules like the quotient rule or product rule. Always check for errors in the calculation and ensure it is properly simplified.
Mistake 4
Not writing Constants and Coefficients
There is a common mistake that students at times forget to multiply the constants placed before f(x). For example, they incorrectly write d/dx (5f(x)) without multiplying by 5. Students should check the constants in the terms and ensure they are multiplied properly.
Mistake 5
Not Applying the Chain Rule
Students often forget to use the chain rule. This happens when the derivative of the inner function is not considered. To fix this error, students should divide the functions into inner and outer parts. Then, make sure that each function is differentiated.
Examples Using the Derivative of f(x)
Problem 1
Calculate the derivative of (f(x)·g(x))
Here, we have h(x) = f(x)·g(x). Using the product rule, h'(x) = u′v + uv′ In the given equation, u = f(x) and v = g(x).
Let’s differentiate each term, u′= d/dx (f(x)) v′= d/dx (g(x))
Substituting into the given equation, h'(x) = (f'(x)).(g(x)) + (f(x)).(g'(x))
Let’s simplify terms to get the final answer, h'(x) = f'(x)g(x) + f(x)g'(x)
Thus, the derivative of the specified function is f'(x)g(x) + f(x)g'(x).
Explanation
We find the derivative of the given function by dividing the function into two parts. The first step is finding its derivative and then combining them using the product rule to get the final result.
Problem 2
A company’s revenue is represented by the function R(x) = f(x), where R represents revenue at a certain production level x. If x = 100 units, measure the rate of change of revenue.
We have R(x) = f(x) (revenue function)...(1)
Now, we will differentiate the equation (1) Take the derivative f(x): dR/dx = f'(x)
Given x = 100, substitute this into the derivative Measure f'(100) to find the rate of change of revenue.
Hence, we get the rate of change of revenue at a production level of x = 100 as f'(100).
Explanation
We find the rate of change of revenue at x = 100 by evaluating the derivative f'(100), which tells us how revenue changes with respect to production level at that specific point.
Problem 3
Derive the second derivative of the function y = f(x).
The first step is to find the first derivative, dy/dx = f'(x)...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [f'(x)]
This gives us the second derivative, which is denoted by f''(x).
Therefore, the second derivative of the function y = f(x) is f''(x).
Explanation
We use the step-by-step process, where we start with the first derivative. We then differentiate the first derivative to find the second derivative.
Problem 4
Prove: d/dx (f(x)²) = 2f(x)f'(x).
Let’s start using the chain rule: Consider y = f(x)² [f(x)]²
To differentiate, we use the chain rule: dy/dx = 2f(x).d/dx [f(x)]
Since the derivative of f(x) is f'(x), dy/dx = 2f(x).f'(x)
Substituting y = f(x)², d/dx (f(x)²) = 2f(x).f'(x)
Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace f(x) with its derivative. As a final step, we substitute y = f(x)² to derive the equation.
Problem 5
Solve: d/dx (f(x)/x)
To differentiate the function, we use the quotient rule: d/dx (f(x)/x) = (d/dx (f(x)).x - f(x).d/dx(x))/x²
We will substitute d/dx (f(x)) = f'(x) and d/dx (x) = 1 (f'(x)x - f(x)·1) / x² = (f'(x)x - f(x)) / x²
Therefore, d/dx (f(x)/x) = (f'(x)x - f(x)) / x²
Explanation
In this process, we differentiate the given function using the quotient rule. As a final step, we simplify the equation to obtain the final result.
FAQs on the Derivative of f(x)
1.Find the derivative of f(x).
2.Can we use the derivative of f(x) in real life?
3.Is it possible to take the derivative of f(x) at points where it is not defined?
4.What rule is used to differentiate f(x)/x?
5.Are the derivatives of f(x) and f⁻¹(x) the same?
Important Glossaries for the Derivative of f(x)
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Differentiability: The property of a function that means it has a defined derivative at all points in its domain.
- Chain Rule: A rule used to differentiate composite functions.
- Product Rule: A rule used to differentiate functions that are products of two or more functions.
- Quotient Rule: A rule used to differentiate functions that are ratios of two functions.
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
