Why are they taking the derivative of 1 - x?

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In summary, the conversation discusses the process of taking the derivative of f(x) = 4 / sqrt(1 - x) and the use of Wolfram Alpha to solve the problem. The individual has trouble understanding why the derivative of 1 - x is taken in the numerator after using the chain rule on 1 / sqrt(1 - x). It is explained that this is because of the use of the chain rule, where d/dx(1/√u) = d/du(1/√u) * du/dx. After some time, the individual comes to understand the reasoning behind this step.
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communitycoll
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Homework Statement


I'm trying to take the derivative of f(x) = 4 / sqrt(1 - x).

Normally if I have trouble I just plug my problem into Wolfram Alpha, and it saves the day.


Homework Equations


However, I don't know why it takes the derivative of 1 - x in the numerator after using the chain rule on 1 / sqrt(1 - x) here:

http://www.wolframalpha.com/input/?i=4+/+sqrt(1-+x)


The Attempt at a Solution


I'll say that I understand everything up until the aforementioned point in the problem.
 
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  • #2
communitycoll said:

Homework Statement


I'm trying to take the derivative of f(x) = 4 / sqrt(1 - x).

Normally if I have trouble I just plug my problem into Wolfram Alpha, and it saves the day.


Homework Equations


However, I don't know why it takes the derivative of 1 - x in the numerator after using the chain rule on 1 / sqrt(1 - x) here:

http://www.wolframalpha.com/input/?i=4+/+sqrt(1-+x)


The Attempt at a Solution


I'll say that I understand everything up until the aforementioned point in the problem.

Because they're using the chain rule.
d/dx(1/√u) = d/du(1/√u) * du/dx
 
  • #3
Ah! It took me awhile, but I finally see what they're doing. Thanks.
 

Related to Why are they taking the derivative of 1 - x?

1. Why do we need to take the derivative of 1 - x?

Taking the derivative of a function helps us understand the rate of change of that function. In this case, we are interested in finding the slope of the line tangent to the graph of 1 - x at any given point. This can be useful in solving problems involving rates of change, optimization, and finding critical points.

2. How do you take the derivative of 1 - x?

To take the derivative of 1 - x, we use the power rule of differentiation. Since 1 can be written as x^0, the derivative of 1 is 0. For x, the derivative is 1. Therefore, the derivative of 1 - x is 0 - 1, which simplifies to -1. So, the derivative of 1 - x is simply -1.

3. What is the significance of taking the derivative of 1 - x?

The derivative of 1 - x allows us to understand the instantaneous rate of change of the function at any given point. It also helps us find the slope of the tangent line, which can be used to solve problems involving optimization, related rates, and curve sketching.

4. Can you explain the geometric interpretation of taking the derivative of 1 - x?

The derivative of 1 - x is the slope of the tangent line to the graph of the function at any given point. This means that the derivative represents the rate of change of 1 - x at that point. Geometrically, it can be thought of as the steepness of the curve at that point.

5. Is taking the derivative of 1 - x useful in real-life applications?

Yes, taking the derivative of 1 - x has many real-life applications. For example, in physics, the derivative can be used to calculate the velocity of an object at a specific time. It is also used in economics and business to analyze rates of change and optimize production processes. In engineering, the derivative can be used to design and optimize structures and systems.

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