Find the value of x *<br> A = 48 in.²

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

tatyana61 [14]

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!

5 0

3 years ago

Steel loading ramps are used to load a

tia_tia [17]

Answer:

6.25 feet.

Step-by-step explanation:

Let L be the length of the ramp in feet.

We have been given that steel loading ramps are used to load a lawn mower onto a truck bed 37.5 inches above the ground. The ramp make a 30° angle with the ground.

We can see from our attachment that ramp and truck bed forms a right triangle with ground. The truck bed is opposite side and length of ramp is hypotenuse of our given angle.

Since we know that Sine relates the opposite and hypotenuse of a right triangle, so we will use Sine to solve for L.

Upon substituting our given values in above formula we will get,

Therefore, the length of the ramp is 75 inches.

Let us convert the length of ramp in feet.

1 feet = 12 inches.

Therefore, the length of the ramp is 6.25 feet.

6 0

3 years ago

How to solve this equation

mariarad [96]

$\bf \sqrt{5x+19}-4=3x-15
\\\\
\sqrt{5x+19}=3x-15+4\\\\ \sqrt{5x+19}=3x-11\leftarrow \textit{square both sides}
\\\\
(\sqrt{5x+19})^2=(3x-11)^2
\\\\
5x+19=(3x-11)^2\qquad 
\begin{cases}
(3x-11)^2\to (3x-11)(3x-11)\\
 --------------\\
\textit{or use FOIL to expand it}
\end{cases}$

then solve for "x"

7 0

3 years ago

36 miles per 6 gallons

stepladder [879]

6 m.p.g if that's the type of question your referring to...

4 0

3 years ago

Read 2 more answers

Are the following equations equivalent why or why not Branilist for people who explain

wariber [46]

Answer:

Yes, they are equivilant

Step-by-step explanation:

5(3m+2p)-4m = 13p+11m-3p

Distribute 5: 15m+10p-4m = 13p+11m-3p

Combine like terms: 15m+(-4m) = 11m ; 13p+(-3p) = 10p

11m+10p = 11m+10p

6 0

3 years ago

Find the value of x * A = 48 in.²