100 LearnersLast updated on July 21st, 2025
Derivative of -3x
We use the derivative of -3x, which is -3, as a tool for understanding how linear functions change in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now discuss the derivative of -3x in detail.
What is the Derivative of -3x?
We now understand the derivative of -3x. It is commonly represented as d/dx (-3x) or (-3x)', and its value is -3. The function -3x has a clearly defined derivative, indicating it is linear and differentiable across its domain. The key concepts are mentioned below: Linear Function: (-3x) is a linear function with a constant slope of -3. Power Rule: A rule used for differentiating functions involving x raised to a power. Constant Rule: A method for finding the derivative of a constant multiplied by a variable.
Derivative of -3x Formula
The derivative of -3x can be denoted as d/dx (-3x) or (-3x)'. The formula we use to differentiate -3x is: d/dx (-3x) = -3 (or) (-3x)' = -3 The formula applies to all x without exception, as -3x is a linear function.
Proofs of the Derivative of -3x
We can derive the derivative of -3x using basic differentiation rules. To show this, we will use the power rule and constant rule. Here are the steps: By Power Rule The derivative of -3x can be proved using the Power Rule, which states that for any function of the form ax^n, the derivative is n*ax^(n-1). Consider f(x) = -3x, where a = -3 and n = 1. f'(x) = 1*(-3)x^(1-1) = -3x^0 = -3, since any number raised to the power of 0 is 1. Hence, proved. Using Constant Rule To prove the differentiation of -3x using the Constant Rule, We use the formula: d/dx (cx) = c, where c is a constant. For -3x, c = -3. Thus, d/dx (-3x) = -3. Hence, proved.
Higher-Order Derivatives of -3x
When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. For a linear function like -3x, the higher-order derivatives simplify quickly. For the first derivative of the function, we write f'(x), which indicates how the function changes or its slope at a certain point. The second derivative of -3x, denoted as f''(x), is 0, since the rate of change of a constant is zero. Similarly, the third derivative, f'''(x), and all subsequent derivatives are also 0. For the nth Derivative of -3x, we generally use f^n(x) for the nth derivative of a function f(x), which tells us the change in the rate of change. For n >= 2, it is 0.
Special Cases:
When differentiating a linear function like -3x, there are no special cases related to undefined points, as the function is continuous and differentiable everywhere. When x is any real number, the derivative of -3x is always -3.
Common Mistakes and How to Avoid Them in Derivatives of -3x
Students frequently make mistakes when differentiating -3x. 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 expression
Students may forget to simplify the expression when differentiating and sometimes incorrectly apply rules meant for non-linear functions. Since -3x is linear, its derivative is simply the constant -3. Ensure that each step is clear and straightforward, utilizing the correct rule.
Mistake 2
Confusing Linear and Non-linear Functions
Some students might confuse linear functions with non-linear ones, leading to incorrect application of rules like the power or product rule unnecessarily. Remember that -3x is linear, and its derivative is straightforward.
Mistake 3
Incorrect use of Rules
While differentiating, students might incorrectly apply rules like the chain rule, which is unnecessary for a linear function. Ensure you apply the constant rule directly: d/dx (cx) = c.
Mistake 4
Ignoring Constants
There is a common mistake where students forget to consider the constant multiplier before x, leading to incorrect results. Always multiply x by its coefficient before differentiating.
Mistake 5
Confusing Higher-Order Derivatives
Students may mistakenly believe that higher-order derivatives of linear functions are non-zero. For -3x, any derivative beyond the first is zero. Recognize that the higher derivatives of linear functions are zero.
Examples Using the Derivative of -3x
Problem 1
Calculate the derivative of -3x·5x
Here, we have f(x) = -3x·5x. Using the product rule, f'(x) = u'v + uv' In the given equation, u = -3x and v = 5x. Let’s differentiate each term, u' = d/dx (-3x) = -3 v' = d/dx (5x) = 5 Substituting into the given equation, f'(x) = (-3)·(5x) + (-3x)·(5) Let’s simplify terms to get the final answer, f'(x) = -15x - 15x = -30x Thus, the derivative of the specified function is -30x.
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 is analyzing its revenue, represented by the function y = -3x, where y represents the loss in dollars for x items produced. If x = 10, calculate the rate of loss.
We have y = -3x (rate of loss)...(1) Now, we will differentiate the equation (1) Take the derivative of -3x: dy/dx = -3 Given x = 10, The rate of loss is constant at -3 for any x, Hence, the rate of loss for producing 10 items is -3 dollars per item.
Explanation
We find the rate of loss at x=10 as -3, which means that for each additional item produced, the company incurs a loss of 3 dollars.
Problem 3
Derive the second derivative of the function y = -3x.
The first step is to find the first derivative, dy/dx = -3...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx (-3) The derivative of a constant is 0, Therefore, the second derivative of the function y = -3x is 0.
Explanation
We use the step-by-step process, where we start with the first derivative. Since the first derivative is a constant, its derivative, which is the second derivative, is 0.
Problem 4
Prove: d/dx (-3x²) = -6x.
Let’s start using the power rule: Consider y = -3x² To differentiate, we use the power rule: dy/dx = -3 * 2x^(2-1) = -6x Hence proved.
Explanation
In this step-by-step process, we used the power rule to differentiate the equation. The power of x is reduced by one, and the coefficient is multiplied by the original power to find the derivative.
Problem 5
Solve: d/dx (-3x/x)
To differentiate the function, we simplify first: -3x/x = -3 The derivative of a constant is 0, Therefore, d/dx (-3x/x) = 0.
Explanation
In this process, we simplify the given function first, recognizing that it reduces to a constant, whose derivative is 0.
FAQs on the Derivative of -3x
1.Find the derivative of -3x.
2.Can we use the derivative of -3x in real life?
3.Is it possible to take the derivative of -3x at any point?
4.What rule is used to differentiate -3x/x?
5.Are the derivatives of -3x and -3/x the same?
Important Glossaries for the Derivative of -3x
Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Linear Function: A function of the form ax + b, which has a constant rate of change. Power Rule: A rule used to differentiate functions of the form ax^n, with the derivative n*ax^(n-1). Constant Rule: A rule stating that the derivative of a constant times a variable is the constant itself. Higher-Order Derivative: Derivatives obtained from differentiating a function multiple times, often leading to zero for linear functions.
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Jaskaran Singh Saluja
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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.
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