1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Fudgin [204]
3 years ago
9

Find dy/dx by implicit differentiation for ysin(y) = xcos(x)

Mathematics
1 answer:
tatyana61 [14]3 years ago
5 0

Answer:

\frac{dy}{dx}=\frac{\cos(x)-x\sin(x)}{\sin(y)+y\cos(y)}

Step-by-step explanation:

So we have:

y\sin(y)=x\cos(x)

And we want to find dy/dx.

So, let's take the derivative of both sides with respect to x:

\frac{d}{dx}[y\sin(y)]=\frac{d}{dx}[x\cos(x)]

Let's do each side individually.

Left Side:

We have:

\frac{d}{dx}[y\sin(y)]

We can use the product rule:

(uv)'=u'v+uv'

So, our derivative is:

=\frac{d}{dx}[y]\sin(y)+y\frac{d}{dx}[\sin(y)]

We must implicitly differentiate for y. This gives us:

=\frac{dy}{dx}\sin(y)+y\frac{d}{dx}[\sin(y)]

For the sin(y), we need to use the chain rule:

u(v(x))'=u'(v(x))\cdot v'(x)

Our u(x) is sin(x) and our v(x) is y. So, u'(x) is cos(x) and v'(x) is dy/dx.

So, our derivative is:

=\frac{dy}{dx}\sin(y)+y(\cos(y)\cdot\frac{dy}{dx}})

Simplify:

=\frac{dy}{dx}\sin(y)+y\cos(y)\cdot\frac{dy}{dx}}

And we are done for the right.

Right Side:

We have:

\frac{d}{dx}[x\cos(x)]

This will be significantly easier since it's just x like normal.

Again, let's use the product rule:

=\frac{d}{dx}[x]\cos(x)+x\frac{d}{dx}[\cos(x)]

Differentiate:

=\cos(x)-x\sin(x)

So, our entire equation is:

=\frac{dy}{dx}\sin(y)+y\cos(y)\cdot\frac{dy}{dx}}=\cos(x)-x\sin(x)

To find our derivative, we need to solve for dy/dx. So, let's factor out a dy/dx from the left. This yields:

\frac{dy}{dx}(\sin(y)+y\cos(y))=\cos(x)-x\sin(x)

Finally, divide everything by the expression inside the parentheses to obtain our derivative:

\frac{dy}{dx}=\frac{\cos(x)-x\sin(x)}{\sin(y)+y\cos(y)}

And we're done!

You might be interested in
Please help, i can't figure this out and im ready to pull my hair out!!!
Nat2105 [25]

Answer:

I'm sorry I don't know the equation, but she walks 3 mph.

Step-by-step explanation:

We know that the answer is asking for speed, which needs both distance and time, so we also need both. Which is where the 9 mph is used. 9 mph means its a difference. b means biking mph, w means walking mph, sooo ----->>> b=w+9, and the difference in the 1st equation is 36, which means the first equation is 4 times larger than the b=w+9, so she walked 48 miles, as well as walked 12 miles in 4 hours. Divide then distances by 4, and you get 12 miles and 3 miles in one hour. Or 12 mph & 3 mph, which fits into 12=3+9. So she walks 3 miles per hour.

5 0
3 years ago
Describe how the graph of y= x2 can be transformed to the graph of the given equation. y = x2 + 3 (5 points)
kari74 [83]

Answer:

x=-1 1/2,x=-1.5

Step-by-step explanation: x=-3/2

6 0
3 years ago
You're having dinner at a restaurant that serves 555 kinds of pasta (spaghetti, bow ties, fettuccine, ravioli, and macaroni) in
bekas [8.4K]

Answer:

Is B i just did it on khan

The answer is B that is beacuse you see the pasta on the side take your time look closly 1 have tomato on Pasta side so B is the only correct answer.

6 0
3 years ago
Read 2 more answers
What are the solutions to the equations |x-10|-4=2x
Tamiku [17]
X is going to equal two in this equation.
7 0
3 years ago
Help please!<br><br> Help would be appreciated.
kotegsom [21]

Answer:

\displaystyle 157,5

Step-by-step explanation:

\displaystyle 3\frac{3}{4}\:quibs = 1\:slep \\ \\ 3\frac{3}{4} \times 42 = 157\frac{1}{2} = 157,5

I am joyous to assist you anytime.

5 0
3 years ago
Other questions:
  • Molten monsters from depths of the earth have surfaced; each has 3 heads and 47 legs.At the first meeting between humans and mon
    5·1 answer
  • An encyclopedia has eight volumes. In how many ways can the eight volumes be replaced on the shelf? Explain.
    15·1 answer
  • Explain how the GCF helps with the distributive property. Why is it so important to use the GCF when factoring a sum of two numb
    11·1 answer
  • Which fraction is equivalent to 2/5?  <br> A. 4/10 <br> B. 2/10   <br> C. 1/2   <br> D. 4/6
    6·1 answer
  • Sorry if it is not clear the question asks, Two squares of side x cm are removed from a square of side 3x cm as shown
    15·1 answer
  • A group of students calculated their average score at a spelling bee. They realized that if one of them scored 9 more points, th
    13·1 answer
  • Which coordinate plane shows the graph of the function y = ?
    10·2 answers
  • Question 1 of 10
    6·2 answers
  • I need help plz 8th math
    9·2 answers
  • Pls help me frens......
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!