Question

Find dy/dx by implicit differentiation
a) x2 + xy - y2 = 7
b) 7sin(x)cos(y) = -2

Answer 1

The
`$(d y)/(d x)$` of
`$x^2+x y-y^2=7$` and
`$7 \sin (x) \cos (y)=-2$` by implicit differentiation are
`(-2 x+y)/(x-2 y)` and
`(\cos (x) \cos (y))/(\sin (x) \sin (y))` respectively.

To, find
`$(d y)/(d x)$` using implicit differentiation for the given equations.

a)
`$x^2+x y-y^2=7$`

To find
`$(d y)/(d x)$`, we'll take the derivative of both sides of the equation with respect to
`$x$` :

`(d)/(d x) \left(x^2+x y-y^2\right)= (d)/(d x) 7`

Using the chain rule and product rule where necessary:

`$$\begin{aligned}& (d)/(d x)\left(x^2\right)+(d)/(d x)(x y)-(d)/(d x)\left(y^2\right)=0 \\& 2 x+x (d y)/(d x)+y-2 y (d y)/(d x)=0\end{aligned}$$`

Now, let's collect terms involving $\frac{d y}{d x}$ on one side:

`$$\begin{aligned}& x (d y)/(d x)-2 y (d y)/(d x)=-2 x+y \\& (d y)/(d x)(x-2 y)=-2 x+y\end{aligned}$$`

Finally, isolate
`$(d y)/(d x)$` :

`(d y)/(d x) = (-2 x+y)/(x-2 y)`

That's the derivative
`$(d y)/(d x)$` expressed in terms of
`$x$` and
`$y$` for the given equation.

b)
`$7 \sin (x) \cos (y)=-2$`

Let's find
`$(d y)/(d x)$` for this equation by implicit differentiation:

`(d)/(d x) (7 \sin (x) \cos (y)) = (d)/(d x)(-2)`

Applying the chain rule and product rule where necessary:

`$$7 \cos (x) \cos (y) (d x)/(d x)-7 \sin (x) \sin (y) (d y)/(d x)=0$$`

Simplify the expression:

`$$7 \cos (x) \cos (y)-7 \sin (x) \sin (y) (d y)/(d x)=0$$`

Now, isolate
`$(d y)/(d x)$` :

`$$\begin{aligned}& 7 \sin (x) \sin (y) (d y)/(d x)=7 \cos (x) \cos (y) \\& (d y)/(d x)=(7 \cos (x) \cos (y))/(7 \sin (x) \sin (y)) \\& (d y)/(d x)=(\cos (x) \cos (y))/(\sin (x) \sin (y))\end{aligned}$$`

Therefore, the derivative
`$(d y)/(d x)$` expressed in terms of
`$x$` and
`$y$` for the given equation.

The complete question is here:

Find
`$\mathrm{dy} / \mathrm{dx}$` by implicit differentiation

a)
`$x^2+x y-y^2=7$`

b)
`$7 \sin (x) \cos (y)=-2$`

Answer 2

The derivatives of the functions are

a) dy/dx = -2x - y/x - 2y

b)dy/dx = -cos(x)cos(y)/sin(x)sin(y)

How to differentiate implicit function

a) Let us use implicit differentiation.

x² + xy - y² = 7

Differentiate both sides of the equation with respect to x.

2x + d/dx(xy) - d/dx(y²) = 0

2x + xdy/dx + y - 2ydy/dx = 0

Now, collect terms involving dy/dx

xdy/dx - 2y dy/dx = -2x - y

dy/dx(x - 2y) = -2x - y

dy/dx = -2x - y/x - 2y

b) For the equation 7sin(x)cos(y) = -2 differentiate both sides with respect to x.

7cos(x)cos(y)dx/dx - 7sin(x)sin(y)dy/dx = 0

Simplify:

7cos(x)cos(y) + 7sin(x)sin(y)dy/dx= 0

Now, solve for dy/dx

7sin(x)sin(y)dy/dx = -7cos(x)cos(y)

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