Find the first partial derivatives of the function. f x, y = x8y4 + 8x9y fx x, y

Question

Find the first partial derivatives of the function. f x, y = x8y4 + 8x9y fx x, y = fy x, y = Find the first partial derivatives of the function. z = Ax + 3y 10 z/ x = z/ y = Find the indicated partial derivatives. f x, y = arctan y/x ; fx 8, -7 fx 8, -7 = Use implicit differentiation to find z/ x and z/ y. x2 + 2y2 + 3z2 = 4 z/ x = z/ y = Find all the second partial derivatives. f x, y = x4y7 + 3x9y fxx x, y = fxy x, y = fyx x, y = fyy x, y = Find the indicated partial derivative. f x, y, z = exyz7; fxyz fxyz x, Y, Z = Find the indicated partial derivative s . f x, y = x9y9 - x8y8; fxxx, fxyx fxxx x, y = fxyx x, y =

Explanation / Answer

1. f(x,y) = x8y4  +8x9y

now for the partial derivative we should consider only one variable as the functioning variable other one should be treated as constant.

so here,

fx(x,y) = 8x7y4 + 72x8y-----------------ans

fy(x,y) = 4x8y3  + 8x9--------------------ans

2. z = (4x + 3y)10

here also do the same, so,

dz/dx = 40(4x + 3y)9

dz/dy = 30(4x + 3y)9-----------------------------ans.

3. f(x,y) = arctan(y/x)

so, fx(x,y) = 1/(x2 +y2) d/dx(y/x) = -y/(x2 +y2)

so, fx(8,-7) = 7/113-----------------ans.

4.

here given that

x2 + 2y2 + 3z2 = 4---

we need to convert it as a function of z so taking z in the left side of the equation and taking square root we can get,

z = (4-x2 - 2y2/3)1/2

now taking the derivative w.r.to x we can have,

dz/dx = - x/(3*(4-x2 - 2y2/3)1/2)--------------------ans

similarly dz/dy = -2y/(3*(4-x2 - 2y2/3)1/2)---------------------ans.

5. the notation and their corresponding meanings are as follows,

fxx = d/dx(df/dx)---------1

fxy = d/dx(df/dy)--------2

fyx = d/dy(df/dx)-------3

fyy =d/dy(df/dy)-------4

so, as dne above we can find,

fxx = 12x2y7 +216x7y

fxy = 28x3y6 + 27x8

fyx = fxy =  28x3y6 + 27x8

fyy = 42 x4y5

6.

here also

fxyz   means d/dx(d/dy(df/dz))

so,

fxyz = d/dx(d/dy)(x*exp(xy)) = d/dx(x*exp(xy) + x2y*exp(xy)) = exp(xy)   + 3yx*exp(xy)   + x2y2*exp(xy) ---------------------ans

7.

similary,

fxxx = d/dx(d/dx(df/dx)) = 504x6y9 - 336x5y8 -----------ans

and

fxyx   = d/dx(d/dy(df/dx)) = 648x7y8 - 448 x6y7 --------------------ans

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