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Chapter 1: Problem 53

Find the indicated derivative. $$\frac{d y}{d x} \text { if } y=1$$

Short Answer

Expert verified
\( \frac{dy}{dx} = 0 \)

Step by step solution

01

Understand the Problem

The task is to find the derivative of the constant function given by the equation y = 1.
02

Recall the Derivative of a Constant

The derivative of a constant function is always zero because the constant does not change as x changes.
03

Apply the Derivative Rule

Using the rule stated in the previous step, write that the derivative of y with respect to x is zero. Mathematically, this is represented as: \[ \frac{dy}{dx} = 0 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant Function
A constant function is a function that always returns the same value, regardless of the input. In mathematical terms, a constant function can be written as:
y = c
where c is some fixed value. In this context, our constant function is y = 1 . This means that no matter what value of x you input, the output will always be 1 . Understanding constant functions is essential because they are the simplest types of functions and form the basis for more complex concepts in calculus. Constant functions are typically graphed as horizontal lines in the Cartesian plane.
Derivative Rule
The derivative rule for a constant function is simple but fundamental. It states that the derivative of a constant function is always zero. This is because a constant function does not change as the variable changes. In mathematical terms:
If \( f(x) = c \), then \( f'(x) = 0 \).
This is a direct consequence of the definition of a derivative, which measures the rate of change of a function. Since a constant function does not change, its rate of change is zero. This rule is highly useful for simplifying more complex differentiation problems.
Differentiation Techniques
Differentiation techniques are methods used to find the derivative of a function. For the problem at hand, we used the most basic technique, which is the derivative of a constant. Here are some other basic techniques:
  • **Power Rule**: If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \)
  • **Sum/Difference Rule**: If \( f(x) = g(x) \text{+/-} h(x) \), then \( f'(x) = g'(x) \text{+/-} h(x) \)
  • **Product Rule**: If \( f(x) = g(x)h(x) \), then \( f'(x) = g'(x)h(x) + g(x)h'(x) \)
  • **Quotient Rule**: If \( f(x) = \frac{g(x)}{h(x)} \), then \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{h(x)^2} \)
  • **Chain Rule**: For a composite function \( f(g(x)) \), the derivative is \( f'(g(x))g'(x) \).
Mastering these techniques allows you to tackle a wide range of differentiation problems. While the differentiation of a constant function is straightforward, these methods provide a toolkit for more complex scenarios.

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Most popular questions from this chapter

(a) Let \(A(x)\) denote the number (in hundreds) of computers sold when \(x\) thousand dollars is spent on advertising. Represent the following statement by equations involving \(A\) or \(A^{\prime}:\) When \(\$ 8000\) is spent on advertising, the number of computers sold is 1200 and is rising at the rate of 50 computers for each \(\$ 1000\) spent on advertising. (b) Estimate the number of computers that will be sold if \(\$ 9000\) is spent on advertising.

Use limits to compute \(f^{\prime}(x) .\) $$f(x)=-1+\frac{2}{x-2}$$

Use limits to compute \(f^{\prime}(x) .\) $$f(x)=-x+11$$

Compute the difference quotient $$\frac{f(x+h)-f(x)}{h}.$$ Simplify your answer as much as possible. $$f(x)=-2 x^{2}+x+3$$

Use limits to compute \(f^{\prime}(x) .\) $$f(x)=x+\frac{1}{x}$$
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