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106 LearnersLast updated on October 5, 2025
Derivative of Graphs
We use derivatives to understand how graphs of functions change in response to slight changes in x. Derivatives help us analyze the behavior of functions, such as finding slopes and identifying concavity. We will now discuss the derivative of graphs in detail.
What is the Derivative of a Graph?
The derivative of a graph represents the rate of change of the function at any given point. It is commonly represented as f'(x) or dy/dx and provides the slope of the tangent line at a specific point on the graph.
The key concepts are mentioned below:
Slope: The steepness or incline of a line.
Tangent Line: A straight line that touches the graph at a single point without crossing it.
Differentiability: A function is differentiable at a point if it has a defined derivative at that point.
Derivative Formula
The derivative of a function f(x) can be denoted as f'(x) or dy/dx.
The basic formula for finding the derivative of a function is: f'(x) = lim(h→0) [f(x + h) - f(x)] / h
This formula applies to all x within the domain of the function where the limit exists.
Proofs of the Derivative of Graphs
We can derive the derivative of graphs using several methods, including:
By First Principles: This involves using the limit definition of the derivative.
Using Chain Rule: This is used when differentiating composite functions. -
Using Product Rule: This is used when differentiating products of functions.
We will now demonstrate the derivation using these methods:
By First Principles
The derivative can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. Consider f(x) = x2. Its derivative can be expressed as the following limit: f'(x) = lim(h→0) [f(x + h) - f(x)] / h = lim(h→0) [(x + h)2 - x2] / h = lim(h→0) [x2 + 2xh + h2 - x2] / h = lim(h→0) [2xh + h2] / h = lim(h→0) [2x + h] = 2x Hence, the derivative of x2 is 2x.
Using Chain Rule
To prove the differentiation of a composite function using the chain rule, consider y = g(f(x)). dy/dx = (dg/df) * (df/dx) For example, if y = (x2 + 1)3, let u = x2 + 1, then y = u3. dy/du = 3u2 and du/dx = 2x dy/dx = 3u2 * 2x = 6(x2 + 1)2 * x
Using Product Rule
We will now prove the derivative using the product rule. Consider y = u(x) * v(x). The product rule states: dy/dx = u'(x) * v(x) + u(x) * v'(x) For example, if y = x * ex, then: u(x) = x and v(x) = ex u'(x) = 1 and v'(x) = ex dy/dx = 1 * ex + x * ex = e^x + x * ex
Higher-Order Derivatives
When a function is differentiated multiple times, the derivatives obtained are referred to as higher-order derivatives.
For example, the second derivative, denoted as f''(x), represents the rate of change of the rate of change (acceleration). Higher-order derivatives help in analyzing the concavity and inflection points of graphs.
Special Cases
For functions with discontinuities or sharp corners, the derivative may be undefined at those points.
For example, the derivative of |x| is undefined at x = 0 because the graph has a sharp corner there.
Common Mistakes and How to Avoid Them in Derivatives of Graphs
Students frequently make mistakes when differentiating graphs. 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.
Ensure that each step is written in order. This helps in avoiding errors in the process.
Mistake 2
Forgetting Undefined Points
Students might not remember that functions can be undefined at certain points, such as sharp corners or vertical asymptotes.
Always consider the domain of the function to understand where it is not continuous.
Mistake 3
Incorrect Use of Differentiation Rules
While differentiating functions, students may misapply rules like the quotient or product rule.
Always write the rules correctly and check for computation errors.
Mistake 4
Neglecting Constants and Coefficients
Students sometimes forget to multiply constants in front of functions.
Always check for constants and ensure they are included in the differentiation process.
Mistake 5
Not Applying the Chain Rule
Students often forget to use the chain rule when necessary.
For composite functions, always divide them into inner and outer parts and differentiate each correctly.
Examples Using the Derivative of Graphs
Problem 1
Calculate the derivative of (x^2 * ln(x))
Here, we have f(x) = x^2 * ln(x). Using the product rule, f'(x) = u'v + uv' In the given equation, u = x^2 and v = ln(x). Let’s differentiate each term, u' = d/dx(x^2) = 2x v' = d/dx(ln(x)) = 1/x Substituting into the given equation, f'(x) = (2x)(ln(x)) + (x^2)(1/x) = 2x ln(x) + x Thus, the derivative of the specified function is 2x ln(x) + 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 tracks its revenue with the function R(x) = x^3 - 3x^2 + 2x. Find the rate of change of revenue when x = 1.
We have R(x) = x^3 - 3x^2 + 2x. Differentiate the function: dR/dx = 3x^2 - 6x + 2 Substitute x = 1 into the derivative: dR/dx = 3(1)^2 - 6(1) + 2 = 3 - 6 + 2 = -1 Hence, the rate of change of revenue when x = 1 is -1.
Explanation
We differentiate the revenue function to find its rate of change.
By substituting x = 1, we calculate the specific rate of change at that point.
Problem 3
Derive the second derivative of the function y = e^x * sin(x).
First, find the first derivative using the product rule: dy/dx = e^x * cos(x) + e^x * sin(x) Now, differentiate again for the second derivative: d^2y/dx^2 = e^x * cos(x) - e^x * sin(x) + e^x * sin(x) + e^x * cos(x) = 2e^x * cos(x) Therefore, the second derivative of the function y = e^x * sin(x) is 2e^x * cos(x).
Explanation
We use the step-by-step process to differentiate the function twice, initially using the product rule and then differentiating the resultant expression again.
Problem 4
Prove: d/dx (x^3) = 3x^2.
Consider y = x^3. Differentiate using the power rule: dy/dx = 3x^(3-1) = 3x^2 Hence proved.
Explanation
In this simple proof, we use the power rule to differentiate x3, confirming that its derivative is 3x2.
Problem 5
Solve: d/dx (x^2/x)
To differentiate the function, simplify first: (x^2/x) = x Then differentiate: d/dx(x) = 1 Therefore, d/dx(x^2/x) = 1.
Explanation
We simplify the function before differentiating, recognizing that x2/x simplifies to x, which differentiates to 1.
FAQs on the Derivative of Graphs
1.Find the derivative of x^2.
2.Can derivatives be used in real life?
3.Is it possible to take the derivative of a function at a point where it is not continuous?
4.What rule is used to differentiate x^2 * ln(x)?
5.Are the derivatives of x^2 and x^3 the same?
Important Glossaries for the Derivative of Graphs
- Derivative: The measure of how a function changes as its input changes.
- Slope: The steepness or incline of a line, often represented as the rate of change.
- Tangent Line: A line that touches a graph at one point, representing the instantaneous rate of change.
- Chain Rule: A formula for computing the derivative of the composition of two or more functions.
- Product Rule: A formula used to find the derivative of the product of two functions.
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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.
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: He loves to play the quiz with kids through algebra to make kids love it.
