How Do You Find the Derivative of This Function?

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In summary: To get this, use the distributive property:\frac{d}{dx}\left(a+b\right)=\frac{d}{dx}\left(a^2+b^2\right)+\frac{d}{dx}\left(2ab+b^2\right)Putting everything together, we get:\left(x+\sin(x)\right)\left(\cos(x)-x\sin(x)+\frac{1}{2\sqrt{x}}\right)-\left(x\cos(x)+\sqrt{x}\right)\left(1+\cos(x)\right)In summary, the
  • #1
riri
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Hello, I am having trouble finding the derivative of this function. Any help would be appreciated!

\(\displaystyle \frac{x\cos(x)+\sqrt{x}}{x+\sin(x)}\)

What I tried was expanding it first to like \(\displaystyle (x+\sin(x)(x\cos(x)+\sqrt{x})'-(x\cos(x)+\sqrt{x}\)
But I ended up with a long weird answer and doesn't seem to work!

Can anyone give ideas for this?
 
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  • #2
Hello, riri! :D

I have moved this thread here to our Calculus forum. While the problem does involve trigonometric functions, it is primarily a problem in differential calculus.

Also, I have edited your post to wrap your $\LaTeX$ code in [MATH][/MATH] tags.

We are given:

\(\displaystyle f(x)=\frac{x\cos(x)+\sqrt{x}}{x+\sin(x)}\)

We have a quotient, so I would suggest applying the quotient rule to get:

\(\displaystyle f'(x)=\frac{\left(x+\sin(x)\right)\left(x\cos(x)+\sqrt{x}\right)^{\prime}-\left(x\cos(x)+\sqrt{x}\right)\left(x+\sin(x)\right)^{\prime}}{\left(x+\sin(x)\right)^2}\)

Can you continue?
 
  • #3
Hi Mark! :D
Thank you, as you can see I'm still pretty new to this website but love it so much!

Yup, so I did use the quotient rule and got up to that part, it's just that I'm stuck on what to do after like:

\(\displaystyle \frac{x+\sin(x)(1(-\sin(x))+{x}^{\frac{-1}{2}}-(x\cos(x)+\sqrt{x}(1+\cos(x)}{x^2+\sin(x)^2}\)

I feel the direction I'm going in is wrong because I got a really long equation thing and it's stuck there... I'm hoping there's a simple way to solve this though!

Thank you!
 
  • #4
Let's first look at the numerator:

\(\displaystyle \left(x+\sin(x)\right)\left(x\cos(x)+\sqrt{x}\right)^{\prime}-\left(x\cos(x)+\sqrt{x}\right)\left(x+\sin(x)\right)^{\prime}\)

Now, let's look at the parts still left to differentiate, beginning with:

\(\displaystyle \left(x\cos(x)+\sqrt{x}\right)^{\prime}\)

For the first term, we will need to use the product rule:

\(\displaystyle \frac{d}{dx}\left(x\cos(x)\right)=x(-\sin(x))+1(\cos(x))=\cos(x)-x\sin(x)\)

And for the second term, we need the power rule:

\(\displaystyle \frac{d}{dx}\left(\sqrt{x}\right)=\frac{d}{dx}\left(x^{\frac{1}{2}}\right)=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}\)

Putting this together, we may write:

\(\displaystyle \left(x\cos(x)+\sqrt{x}\right)^{\prime}=\cos(x)-x\sin(x)+\frac{1}{2\sqrt{x}}\)

The other part still left to differentiate is:

\(\displaystyle \left(x+\sin(x)\right)^{\prime}=\frac{d}{dx}\left(x+\sin(x)\right)=1+\cos(x)\)

And so the numerator then becomes:

\(\displaystyle \left(x+\sin(x)\right)\left(\cos(x)-x\sin(x)+\frac{1}{2\sqrt{x}}\right)-\left(x\cos(x)+\sqrt{x}\right)\left(1+\cos(x)\right)\)

You could multiply everything out and see if you can combine terms and use trig. identities to simplify.

For the denominator, you have committed an error so commonly made, it has a special name, which is "The Freshman's Dream." You have stated essentially:

\(\displaystyle (a+b)^2=a^2+b^2\)

This is not true in general, what you want, if you are going to expand it, is:

\(\displaystyle (a+b)^2=a^2+2ab+b^2\)
 

FAQ: How Do You Find the Derivative of This Function?

What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a given function. It measures how much a function changes with respect to its input variable. In other words, it shows how the output of a function changes when the input value changes.

How do you find the derivative of a function?

To find the derivative of a function, you can use the derivative rules which include the power rule, product rule, quotient rule, and chain rule. These rules help you find the derivative of a function by manipulating its algebraic expression.

Why is finding the derivative important in science?

The derivative is an essential tool in science because it helps us understand the behavior and relationships between different variables in a function. It is particularly useful in physics, engineering, and economics, where many physical quantities can be described by mathematical functions.

Can the derivative of a function be negative?

Yes, the derivative of a function can be negative. This indicates that the function is decreasing at that particular point. A positive derivative indicates that the function is increasing at that point, and a zero derivative indicates that the function is not changing at that point.

What is the difference between a derivative and an antiderivative?

A derivative is the rate of change of a function, while an antiderivative is the inverse operation of a derivative. In other words, the derivative of a function gives you the rate of change of that function, while the antiderivative gives you the original function of which the derivative is taken.

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