Question

Problem 1.1.4. (i) Let $f(x, y, z) = xy + xy^2 - \ln(xyz)$. \\ compute the partial derivatives $f_x, f_y, f_z, f_{xy}, f_{xz}$ and $f_{xx}$. Solution: (ii) Let $g(x, y) = xe^y$, compute $g_{xx}(1, 0), g_{xy}(1, 0)$ and $g_{yy}(1, 0)$ Solution: (iii) Let $f(x, y, u, v) = \frac{x^2 + e^{x^2}}{3y^2 + \ln(2 + v^2)}$ What is the fastest way to show that $f_{uvxy}(x, y, u, v) = 0$ for all $(x, y, u, v)$? Solution

          Problem 1.1.4. (i) Let $f(x, y, z) = xy + xy^2 - \ln(xyz)$. \\
compute the partial derivatives $f_x, f_y, f_z, f_{xy}, f_{xz}$ and $f_{xx}$.
Solution:
(ii) Let $g(x, y) = xe^y$, compute $g_{xx}(1, 0), g_{xy}(1, 0)$ and $g_{yy}(1, 0)$
Solution:
(iii) Let
$f(x, y, u, v) = \frac{x^2 + e^{x^2}}{3y^2 + \ln(2 + v^2)}$
What is the fastest way to show that $f_{uvxy}(x, y, u, v) = 0$ for all $(x, y, u, v)$?
Solution

Problem 1.1.4. (i) Let f(x, y, z) = xy + xy^2 - ln(xyz).

compute the partial derivatives fx, fy, fz, fxy, fxz and fxx.
Solution:
(ii) Let g(x, y) = xe^y, compute gxx(1, 0), gxy(1, 0) and gyy(1, 0)
Solution:
(iii) Let
f(x, y, u, v) = (x^2 + e^x^2)/(3y^2 + ln(2 + v^2))
What is the fastest way to show that fuvxy(x, y, u, v) = 0 for all (x, y, u, v)?
Solution

Added by Anthony A.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Joseph David

Step 1

∂g/∂y: Using the chain rule, we have: ∂g/∂y = ∂g/∂(y^2 + 1) * ∂(y^2 + 1)/∂y = z * (1/2)(y^2 + 1)^(-1/2) * 2y = zy / √(y^2 + 1) ∂g/∂z: Since g is directly dependent on z, the partial derivative with respect to z is simply 1: Show more…

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Problem 1.1.4.i Let f(z) = y + âˆš(y^2 + 1). Compute the partial derivatives âˆ‚f/âˆ‚y and âˆ‚f/âˆ‚z. Solution: i) Let g(y) = zâˆš(y^2 + 1). Compute âˆ‚g/âˆ‚y, âˆ‚g/âˆ‚z, and âˆ‚g/âˆ‚z. ii) Let P = e. What is the fastest way to show that f(y, z) = 0 for all y and z?