Question

Let u = e^qr sin-1p, p = sinx, q = z²ln y, r = 1 / z. Find the derivatives au / ax, au / ay, and au / az as a function of x y and z, using the chain rule and expressing u in terms of x, y, and z before derivatives. Then compute the derivatives du / dx, du / dy and du / dz at the point (x, y, z) = (π / 4,1 / 2, -1/2)

Answer 1

Step-by-step explanation:

$u = {e}^{qr} \sin {}^{ - 1} (p) \\ \sin u = {e}^{qr} \sin \: x \\ \cos u \: au = \cos x \: ax \\ \frac{au}{ax} = \frac{ \cos(x) }{ \cos(u) }$

$u = {e}^{qr} { \sin }^{ - 1} (p) \\ u = {e}^{ {z}^{2} ln(y) . \frac{1}{z} } { \sin }^{ - 1} (p) \\ u = {e}^{z ln(y) } { \sin }^{ - 1} (p) \\ \sin(u) = {e}^{z ln(y) } .p \\ ln( \sin(u) ) = z ln(y) \\ \frac{ \cos(u) }{ \sin(u) } au = \frac{z}{y} ay \\ \frac{au}{ay} = \frac{z \sin(u) }{y \cos(u) } \\ \frac{au}{ay} = \frac{z \tan(u) }{y}$

au/az is got from the same expression.