Given $f(x)=3x^2+3x$, Find $\frac{f(x+h)-f(x)}{h}$.A. $6x+3$ B. $6xh+3h+3x$ C. $6xh+3h^2+3h$ D. $6x+3h+3$ E. None Of These
Introduction
In this article, we will explore the concept of finding the derivative of a function using the definition. The derivative of a function is a measure of how the function changes as its input changes. It is a fundamental concept in calculus and has numerous applications in various fields such as physics, engineering, and economics.
The Definition of a Derivative
The derivative of a function f(x) is defined as:
f'(x) = lim(h → 0) [f(x + h) - f(x)]/h
where h is an infinitesimally small change in x.
Given Function
We are given the function f(x) = 3x^2 + 3x. Our goal is to find the derivative of this function using the definition.
Step 1: Evaluate f(x + h)
To find the derivative, we need to evaluate f(x + h). We substitute (x + h) into the function f(x) = 3x^2 + 3x.
f(x + h) = 3(x + h)^2 + 3(x + h)
Expanding the squared term, we get:
f(x + h) = 3(x^2 + 2hx + h^2) + 3x + 3h
Simplifying the expression, we get:
f(x + h) = 3x^2 + 6hx + 3h^2 + 3x + 3h
Step 2: Evaluate f(x + h) - f(x)
Now, we need to evaluate f(x + h) - f(x). We substitute the expressions for f(x + h) and f(x) into this equation.
f(x + h) - f(x) = (3x^2 + 6hx + 3h^2 + 3x + 3h) - (3x^2 + 3x)
Simplifying the expression, we get:
f(x + h) - f(x) = 6hx + 3h^2 + 3h
Step 3: Evaluate [f(x + h) - f(x)]/h
Now, we need to evaluate [f(x + h) - f(x)]/h. We substitute the expression for f(x + h) - f(x) into this equation.
[f(x + h) - f(x)]/h = (6hx + 3h^2 + 3h)/h
Simplifying the expression, we get:
[f(x + h) - f(x)]/h = 6x + 3h + 3
Conclusion
In this article, we used the definition of a derivative to find the derivative of the function f(x) = 3x^2 + 3x. We evaluated f(x + h), f(x + h) - f(x), and [f(x + h) - f(x)]/h, and finally obtained the derivative f'(x) = 6x + 3h + 3.
Answer
The correct answer is D. 6x + 3h + 3.
Discussion
This problem is a great example of how the definition of a derivative can be used to find the derivative of a function. The definition provides a clear and concise way to calculate the derivative, and it is a fundamental concept in calculus.
In this problem, we used the definition of a derivative to find the derivative of the function f(x) = 3x^2 + 3x. We evaluated f(x + h), f(x + h) - f(x), and [f(x + h) - f(x)]/h, and finally obtained the derivative f'(x) = 6x + 3h + 3.
This problem is a great example of how the definition of a derivative can be used to find the derivative of a function. The definition provides a clear and concise way to calculate the derivative, and it is a fundamental concept in calculus.
Related Problems
- Find the derivative of the function f(x) = 2x^3 - 5x^2 + 3x - 1 using the definition.
- Find the derivative of the function f(x) = x^2 + 2x - 1 using the definition.
- Find the derivative of the function f(x) = 3x^2 + 2x - 1 using the definition.
Solutions
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Find the derivative of the function f(x) = 2x^3 - 5x^2 + 3x - 1 using the definition. f'(x) = lim(h → 0) [f(x + h) - f(x)]/h f'(x) = lim(h → 0) [(2(x + h)^3 - 5(x + h)^2 + 3(x + h) - 1) - (2x^3 - 5x^2 + 3x - 1)]/h f'(x) = lim(h → 0) [6x^2h + 6xh^2 + 3h^2 + 3h]/h f'(x) = 6x^2 + 6x + 3
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Find the derivative of the function f(x) = x^2 + 2x - 1 using the definition. f'(x) = lim(h → 0) [f(x + h) - f(x)]/h f'(x) = lim(h → 0) [(x + h)^2 + 2(x + h) - 1 - (x^2 + 2x - 1)]/h f'(x) = lim(h → 0) [2xh + h^2 + 2h]/h f'(x) = 2x + 2
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Find the derivative of the function f(x) = 3x^2 + 2x - 1 using the definition. f'(x) = lim(h → 0) [f(x + h) - f(x)]/h f'(x) = lim(h → 0) [(3(x + h)^2 + 2(x + h) - 1) - (3x^2 + 2x - 1)]/h f'(x) = lim(h → 0) [6xh + 3h^2 + 2h]/h f'(x) = 6x + 3
Q&A: Finding the Derivative of a Function using the Definition ===========================================================
Q: What is the definition of a derivative?
A: The derivative of a function f(x) is defined as:
f'(x) = lim(h → 0) [f(x + h) - f(x)]/h
where h is an infinitesimally small change in x.
Q: How do I find the derivative of a function using the definition?
A: To find the derivative of a function using the definition, you need to follow these steps:
- Evaluate f(x + h) by substituting (x + h) into the function f(x).
- Evaluate f(x + h) - f(x) by subtracting f(x) from f(x + h).
- Evaluate [f(x + h) - f(x)]/h by dividing f(x + h) - f(x) by h.
- Take the limit as h approaches 0 of [f(x + h) - f(x)]/h.
Q: What is the significance of the limit as h approaches 0?
A: The limit as h approaches 0 is used to define the derivative because it represents the instantaneous rate of change of the function at a given point. As h approaches 0, the difference quotient [f(x + h) - f(x)]/h approaches the instantaneous rate of change of the function at x.
Q: Can I use the definition of a derivative to find the derivative of any function?
A: Yes, you can use the definition of a derivative to find the derivative of any function. However, the definition may become complicated for more complex functions, and it may be easier to use other methods such as the power rule or the product rule to find the derivative.
Q: What are some common mistakes to avoid when using the definition of a derivative?
A: Some common mistakes to avoid when using the definition of a derivative include:
- Not evaluating f(x + h) correctly
- Not evaluating f(x + h) - f(x) correctly
- Not evaluating [f(x + h) - f(x)]/h correctly
- Not taking the limit as h approaches 0 correctly
Q: Can I use the definition of a derivative to find the derivative of a function with multiple variables?
A: Yes, you can use the definition of a derivative to find the derivative of a function with multiple variables. However, the definition may become complicated, and it may be easier to use other methods such as the chain rule or the partial derivative rule to find the derivative.
Q: What are some real-world applications of the definition of a derivative?
A: The definition of a derivative has numerous real-world applications, including:
- Physics: The definition of a derivative is used to describe the motion of objects and the forces acting on them.
- Engineering: The definition of a derivative is used to design and optimize systems, such as electronic circuits and mechanical systems.
- Economics: The definition of a derivative is used to model the behavior of economic systems and to make predictions about future economic trends.
Q: Can I use the definition of a derivative to find the derivative of a function with a complex domain?
A: Yes, you can use the definition of a derivative to find the derivative of a function with a complex domain. However, the definition may become complicated, and it may be easier to use other methods such as the Cauchy-Riemann equations or the Wirtinger calculus to find the derivative.
Q: What are some common functions that can be differentiated using the definition of a derivative?
A: Some common functions that can be differentiated using the definition of a derivative include:
- Polynomials: f(x) = ax^n + bx^(n-1) + ... + c
- Rational functions: f(x) = p(x)/q(x)
- Trigonometric functions: f(x) = sin(x), f(x) = cos(x), f(x) = tan(x)
- Exponential functions: f(x) = e^x
Q: Can I use the definition of a derivative to find the derivative of a function with a discontinuity?
A: No, you cannot use the definition of a derivative to find the derivative of a function with a discontinuity. The definition of a derivative requires that the function be continuous at the point of differentiation. If the function has a discontinuity at the point of differentiation, the definition of a derivative is not applicable.