# Finding the differential of the function

In summary, the Mean Value Theorem is satisfied and so we can solve for c using the formula f'c = f(4)-f(1) / 4-1 = 1/9.
We just finished linear approximations and differentials in our course and was hoping I could get some feedback on this question of mine. The following question I have is asking to find the differential of the function:

y= cos(pi)x

We know from the formula

dy = f'(x)*dx

However, we are supposed to use our product rule here to determine the derivative of cos(pi)x.

I came up with -(pi)sin(pi)x but I'm not sure how to solve from that using the rule. The rest is fairly straight forward because it's just putting it back into the formula. Thanks guys for any help.

$$y = \cos(\pi) x$$

$$\frac{dy}{dx} = \cos (\pi)$$$$y = \cos(\pi x)$$

$$\frac{dy}{dx} = -\pi \sin \pi x$$. You are using the chain rule.

$$dy = -\pi \sin \pi x \ dx$$

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I think loadsy means cos(pi x).

In which case it would simply be:

y = cos(pi*x)
dy/dx = -pi*sin(pi*x)

Differential form would be:

dy = -pi*sin(pi*x)*dx

Forget formulas like that. Just remember to multiply both sides by dx after taking the derivative of the function.

Actually it's just written as y = cos pi x. Which I think is basically the same as $$y = \cos\pi x$$

But I understand what you are saying anyways. :D

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Yeah, so go with my answer.

Gotcha! Alright thanks for the help guys. The question was easier than I made it out to be.

Alright I hope you guys don't mind, I just have another question to ask you, but it should go by fast because I know the steps in solving it anyways.

This is under the "Mean Value Theorem" section, and asks:

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers cs that satisfy the conclusion of the Mean Value Theorem.

f(x) = x / x+2 [1,4]

We know the following:

f(x) is differentiable on (1,4) and that (1,4) is part of domain of definition of f'(x).
f(x) is continuous on [1,4] because it is a rational function.

Then we have to solve for f'(x) which I found to be:

f'(x) = x+2*1 - x*2 / (x+2)^2 = -2x / (x+2) but I'm not sure if this is the right derivative for this case. Once I have that the rest I can do to solve for c.

Because from there you'd solve using the formula f'c = f(4)-f(1) / 4-1 = 1/9 after finding out:
f(4) = 4/4+2 = 2/3 and f(1) = 1/1+2 = 1/3

So in summation: all I'm looking to find the answer of, is the derivative of x / x+2

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## 1. What does finding the differential of a function mean?

Finding the differential of a function refers to the process of determining the change in the output of a function for a given change in the input. It involves finding the derivative of the function at a specific point.

## 2. How is the differential of a function different from the derivative?

The derivative of a function represents the instantaneous rate of change at a specific point, whereas the differential represents the actual change in the output of the function for a given change in the input.

## 3. Why is finding the differential of a function important?

Knowing the differential of a function can help in understanding the behavior of the function and predicting its output for different inputs. It also plays a crucial role in optimization and optimization techniques.

## 4. What is the process for finding the differential of a function?

The process for finding the differential of a function involves taking the derivative of the function using the rules of differentiation and then substituting the value of the input for which the differential is being calculated.

## 5. Can the differential of a function be negative?

Yes, the differential of a function can be negative. This indicates that the output of the function is decreasing for a given increase in the input. It can also indicate a decreasing slope for the function at a specific point.

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