# Find dy/dx by implicit differetiation HELP IMMEDIATLY

of 4 cos x sin y =1
here's what i did

d/dx 4 cosx sin y)= 1 d/dx

4sinx cosy=0

y'= 4 sinx cosy

but the answer is tan x tan y

## Answers and Replies

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do ihave to use the power rule and the chain rule in this problem

well what do i have to do may u show me the steps or correct me,

hage567
Homework Helper
I think you need to use the product rule since it is the product of the two functions (cosx and siny).

ok ill try it out

here's what i did 4 d/dx (Cos x) + cos x * d/dx (4) * sin y dy/ dx

is this right or am i wrong with the product rule of this

then i got 4 sin X + sin x * cos y dy/dx

wheres the help man

hage567
Homework Helper
here's what i did 4 d/dx (Cos x) + cos x * d/dx (4) * sin y dy/ dx

is this right or am i wrong with the product rule of this
No, this doesn't look right. If you are going to include the 4 in the product rule, you will have to apply it twice since that way you technically have three terms.

then i got 4 sin X + sin x * cos y dy/dx
In the second term why did the cosx turn into a sinx? You already took the derivative of that in the first term, so the rule says you leave it alone in the second.

OK make sure you understand the product rule. Don't concentrate on the 4, it is just a constant. You could just take it out for now. Try breaking it up like (cosx)(siny) as your two terms. Now try it again.

so like this ( d/dx cos X + sin X d/dx) sin y dy/dx

I mean 4 sin X + cos X * siny dy/dx

then i change it to sin y dy/dx= -4 sin x / cos x

wait a minute so 4 is left alone, so it's like this cos d/dx (x) + (x) d/dx (cos) * siny dy/dx then i change it too

sin y dy/dx= -4cos - sin x

so 4 = 0 right because the rule says d/dx (C) =0

so then this is sin y dy/dx= cos - sin x
then its dy/dx= cos- sin x/ sin y

do i have to use the quotient rule on this part

I think that you are misunderstanding the product rule. Firstly, since 4 is a constant, you can take it out of the situation. So
$$\frac{d}{dx}4cosxsiny$$ = $$4 \frac{d}{dx}cosxsiny$$

Now, the product rule is $$\frac{d}{dx}[f(x)g(x)]=f(x)g'(x) + g(x)f'(x)$$ I can demonstrate the proof if you wish but I won't clutter up the post unnecesarily.

I am still editing this, but I see that you are still posting.. PLEASE use the edit button..... I am still editing this post so check back in a few minutes...

Okay, anyways... So I will do an example problem and show you how to do it.

Let me explain implicit differentiation for you. When you are differentiating terms involving x alone, you can differentiate as usual. However, when you differentiate terms involving y, you must apply the Chain Rule, because you are assuming that y is defined implicitly as a differentiable function of x.

Lets say the problem was:

find dy/dx given that $$y^{3}+y^{2}-5y-x^{2}=-4$$

Solution:

1.Differentiate both sides of the equation with respect to x.

$$\frac{d}{dx}[y^{3}+y^{2}-5y-x^{2}]=\frac{d}{dx}[-4]$$
$$\frac{d}{dx}[y^{3}] + \frac{d}{dx}[y^{2}] - \frac{d}{dx}[5y] - \frac{d}{dx}[x^{2}] = \frac{d}{dx}[-4]$$
$$3y^{2}\frac{dy}{dx}+2y\frac{dy}{dx}-5\frac{dy}{dx}-2x=0$$ (chain rule)

2. Collect the dy/dx terms on the left side of the equation.

$$3y^{2}\frac{dy}{dx}+2y\frac{dy}{dx}-5\frac{dy}{dx}=2x$$

3.Factor dy/dx out of the left side of the equation

$$\frac{dy}{dx}(3y^{2}+2y-5)=2x$$

4.Solve for dy/dx by dividing by $$3y^{2}+2y-5$$

$$\frac{dy}{dx}=\frac{2x}{3y^{2}+2y-5}$$

There ya go... Do you understand now?

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ok can u demonstrate it too me

is it like this cos x d/dx (sin y) dy/dx + siny d/dx cos X

which equals cos x (cos y dy/dx) + siny d /dx (sin X)

Ok, I don't really see where you are getting the dy/dx in the first post... Can you explain it to me? You have d/dx in front of siny, so you haven't differentiated it yet, why do you have dy/dx there?

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then i did this - cos x/ sin y - sin x/ cos y = tan x tan y

I think that you are misunderstanding the product rule. Firstly, since 4 is a constant, you can take it out of the situation. So
$$\frac{d}{dx}4cosxsiny$$ = [/tex] 4 \frac{d}{dx}cosxsiny[/tex]

Now, the product rule is $$\frac{d}{dx}[f(x)g(x)]=f(x)g'(x) + g(x)f'(x)$$ I can demonstrate the proof if you wish but I won't clutter up the post unnecesarily.

I am still editing this, but I see that you are still posting.. PLEASE use the edit button..... I am still editing this post so check back in a few minutes...

Okay, anyways... So I will do an example problem and show you how to do it.

Let me explain implicit differentiation for you. When you are differentiating terms involving x alone, you can differentiate as usual. However, when you differentiate terms involving y, you must apply the Chain Rule, because you are assuming that y is defined implicitly as a differentiable function of x.

Lets say the problem was:

find $$\frac{dy}{dx}$$ given that $$y^{3}+y^{2}-5^{y}-x^{2}=-4$$

Solution:

1.Differentiate both sides of the equation with respect to x.

$$\frac{d}{dx}[y^{3}+y^{2}-5y-x^{2}]=\frac{d}{dx}[-4]$$
$$\frac{d}{dx}[y^{3}] + \frac{d}{dx}[y^{2}] - \frac{d}{dx}[5y] - \frac{d}{dx}[x^{2}] = \frac{d}{dx}[-4]$$
$$3y^{2}\frac{dy}{dx}+2y\frac{dy}{dx}-5\frac{dy}{dx}-2x=0$$ (chain rule)

2. Collect the dy/dx terms on the left side of the equation.

$$3y^{2}\frac{dy}{dx}+2y\frac{dy}{dx}-5\frac{dy}{dx}=2x$$

3.Factor dy/dx out of the left side of the equation

$$\frac{dy}{dx}(3y^{2}+2y-5)=2x$$

4.Solve for dy/dx by dividing by $$3y^{2}+2y-5$$

$$\frac{dy}{dx}=\frac{2x}{3y^{2}+2y-5}$$

There ya go... Do you understand now?
i understand that ok so i do this

d/dx (4 cos x * sin y)= 1

d/dx 4 * d/dx cos X * d/dx sin y = d/dx (1)

d/dx 0 * d/dx sin x * d/dx cos y =0

then d/dx( sinx * cosy= 0

then i did this - cos x/ sin y - sin x/ cos y = tan x tan y
You violated a ton of mathematical rules here, like getting multiplication out of addition.

Remember the product rule is $$\frac{d}{dx}u*v = u \frac{d}{dx}v + v \frac{d}{dx}u = uv' + vu'$$

Now just apply this to your implicit differentiation problem where u = cosx and v = siny.

d/dx (4 cos x * sin y)= 1

d/dx 4 * d/dx cos X * d/dx sin y = d/dx (1)

d/dx 0 * d/dx sin x * d/dx cos y =0

then d/dx( sinx * cosy= 0
Firstly, d/dx (4 cos x * sin y)= 1 is not true... (why?)

Secondly, d/dx 4 * d/dx cos X * d/dx sin y = d/dx (1), you are not applying the product rule correctly. Nor the constant multiple rule.

I gotta go but if you want to survive, you have to review... lots. Do you want me to do an example with the product rule before I go?