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Magazine Chapter 3: Problem 1
Let \(f(x)=7 .\) Using the definition of the derivative, show that \(f^{\prime}(x)=0\) for all values of \(x\)
Short Answer
Expert verified
The derivative \(f'(x) = 0\) for all \(x\).
Step by step solution
01
- Understand the Definition of the Derivative
The derivative of a function at a point is defined by the limit expression: \[f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h}\]This formula expresses how the function changes at any given point and is the slope of the tangent line at that point.
02
- Substitute the Function into the Derivative Definition
The given function is constant: \(f(x) = 7\). Substitute \(f(x + h)\) and \(f(x)\) into the expression: \[f'(x) = \lim_{{h \to 0}} \frac{{7 - 7}}{h}\]
03
- Simplify the Expression
Simplify the expression inside the limit:\[\frac{{7 - 7}}{h} = \frac{0}{h} = 0\]Since the numerator is zero, the entire fraction simplifies to zero.
04
- Evaluate the Limit
Evaluate the limit:\[\lim_{{h \to 0}} 0 = 0\]Since the expression is constant zero, the limit is also zero.
05
- Conclusion
The derivative of the function \(f(x) = 7\) with respect to \(x\) is zero: \[f'(x) = 0\]This result holds for all values of \(x\) because a constant function has a horizontal tangent line and thus a slope of zero everywhere.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Calculus
Calculus is a branch of mathematics focused on changes and motion. It consists of two main parts: differential calculus and integral calculus. Differential calculus, which we're diving into in this article, deals with finding the derivative of a function. It's all about understanding how a function changes. In simple terms, calculus allows us to accurately find rates of change, such as how fast something is moving at any point in time.
To put it simply:
To put it simply:
- Calculus helps us understand change.
- Differential calculus focuses on the rate of change.
- The main tool here is the derivative.
Definition of the Derivative
The derivative of a function gives us the exact rate at which the function's value changes as its input changes. We find this derivative using a limit-based formula: \[f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h}\] This formula might look a bit tricky at first, but it's simply a way to capture how the output of a function shifts with an infinitesimally small change in the input.
For a constant function, such as \(f(x) = 7\), the change is actually zero, because there is no difference between \(f(x + h)\) and \(f(x)\). That's why the derivative results in zero, reflecting that there鈥檚 no change.
For a constant function, such as \(f(x) = 7\), the change is actually zero, because there is no difference between \(f(x + h)\) and \(f(x)\). That's why the derivative results in zero, reflecting that there鈥檚 no change.
- The derivative tells us how fast the function's value is changing.
- With constant functions, the derivative is always zero because there's no change regardless of the input.
Slope of Tangent Line
The tangent line to a function at any point provides an approximation to the function鈥檚 behavior right at that point. Imagine this line just touching the function at a point and not crossing it.
The slope of this tangent line is the derivative at that point. For a constant function like \(f(x) = 7\), any tangent line is perfectly horizontal. This is reflected by the derivative being zero. A horizontal line has no rise, only a run, hence no slope.
The slope of this tangent line is the derivative at that point. For a constant function like \(f(x) = 7\), any tangent line is perfectly horizontal. This is reflected by the derivative being zero. A horizontal line has no rise, only a run, hence no slope.
- A tangent line touches the curve at one exact point.
- In constant functions, the slope of this line is zero.
Limit Process
The limit process is key to finding derivatives. It involves seeing how a function behaves as you get infinitely close to a point. In our derivative formula, the limit \[\lim_{{h \to 0}}\] means we are looking at what happens as \(h\), a small change in input, approaches zero.
This is crucial because we want to capture precise behavior right at a point, not over a range.
This is crucial because we want to capture precise behavior right at a point, not over a range.
- The limit helps us infinitely zoom into a curve at a point.
- For constant functions, the behavior remains the same as we approach any point.
