Derivation of basic functions After observing y '= 2x y = x ^ (1 / 2) y' = 1 / [2x ^ (1 / 2)] y = x ^ 3-4x y '= 3x ^ 2-4 of y = x ^ 2, I guess that for y = x ^ m, the derivative y' of the function in the form of (M > 1) satisfies y '= MX ^ (m-1). Is my guess correct

Derivation of basic functions After observing y '= 2x y = x ^ (1 / 2) y' = 1 / [2x ^ (1 / 2)] y = x ^ 3-4x y '= 3x ^ 2-4 of y = x ^ 2, I guess that for y = x ^ m, the derivative y' of the function in the form of (M > 1) satisfies y '= MX ^ (m-1). Is my guess correct

In fact, (x ^ m) '= MX ^ (m-1) holds for any real number M. it can be proved by the derivation of composite function: because f (x) = x ^ m = e ^ (mlnx), f' (x) = [e ^ (mlnx)] '= [e ^ (mlnx)] * (M / x) =

Find a problem of derivative high number, let the equation x = y Λ y, determine that y is a function of X, and find dy

ylny = lnx
Derivation of X on both sides:
y'lny + y(1/y)y' = 1/x
y'lny + y' = 1/x
(lny + 1)y' = 1/x
y' = 1/[x(lny + 1)]

Let f be a differentiable function, z = Z (x, y) is an implicit function determined by equation y + Z = XF (Y Λ 2-z Λ 2), and prove X σ z/ σ x-z σ z/ σ y=y

It is proved that z = Z (x, y) is composed of equation y + Z = XF (y) ²- z ²) The determined implicit function, so both sides simultaneously derive x with ∂ Z / ∂ x = f (y) ²- z ²)- 2xzf'(y ²- z ²) ∂z/∂x=(y+z)/x-2xzf'(y ²- z ²) ∂z...

Let the function z = Z (x, y) be determined by the equation x ^ 2 + y ^ 2 + Z ^ 2 = XF (Y / x), and f be differentiable, and the partial derivative of Z to x, y

Let f (x) = x ^ 2 + y ^ 2 + Z ^ 2 XF (Y / x) = 0 = x ^ 2 + y ^ 2 + Z ^ 2 XF (U) = 0 u = Y / X ə u/ ə x=-y/x^2=-u/x, ə u/ ə y=1/x ə F/ ə x=2x-f(u)-x* ə f/ ə u* ə u/ ə x=2x-f(u)+ ə f/ ə u*u ə F/ ə y=...

4 "University multivariate function calculus" questions! 1. Find the reciprocal or partial derivative Let x = e ^ u + usinv, y = e ^ u ucosv, find (partial U / partial x), (partial U / partial y), (partial V / partial x), (partial V / partial y) [e ^ u represents the U power of E, the same below] 2. Find the tangent and normal plane equation of the curve X = t / (1 + T), y = (T + 1) / T, z = T ^ 2 at the point corresponding to t = 1 3. Find the tangent and normal plane equation of the space curve at the specified point X ^ 2 + y ^ 2 + Z ^ 2 = 6, x + y + Z = 0, at point (1, - 2,1) 4. Find the directional derivative of the function z = in (x + y) at the point (1,2) on the parabola y = 4x along the tangent direction of the parabola to the positive x-axis [the two equations of questions 1 and 3 are enclosed in curly brackets. In fact, they are not difficult, but I am always different from the standard answer,]

(1) : x = e ^ u + usinv, y = e ^ u-ucosv first calculate the partial derivative of X at the same time to obtain 1 = e ^ u partial U / partial x + SINV partial U / partial x + ucosv partial V / partial x0 = e ^ u partial U / partial xcosv partial U / partial x + usinv partial V / partial X. the partial U / partial x = SINV / [e ^ u (SINV + cosv) + 1], partial V / partial x = (sinv-e ^ u) / [UE ^ u (SINV + CO

Extreme value problem of calculus 1 multivariate function Find the maximum and minimum value of Z = 2x square - 8x-2y + 9 on D: 2x square + y square ≤ 1. It seems that the following boundary value is not easy to find,

Rewrite Z as Z = 2 (X-2) ^ 2-2y + 1 to get y = (X-2) ^ 2 + (1-z) / 2
Find the range of Z, that is, find the range where the parabola y = (X-2) ^ 2 can translate within a given elliptical range, that is, the range of (1-z) / 2
It is not difficult to find from the geometric image that when the maximum and minimum values of (1-z) / 2 are taken, the parabola y = (X-2) ^ 2 + (1-z) / 2 is tangent to the ellipse, and the slope at the tangent point is equal, so let's use the derivative below
Parabola dy / DX = 2 (X-2) (1)
Ellipse 4x + 2Y * dy / DX = 0 dy / DX = - 2x / y = - 2x / ±√ 1-2x ^ 2 (2)
Simultaneous (1) and (2) get 2 (X-2) = - 2x / ±√ 1-2x ^ 2
X ^ 4-4x ^ 3 + 4x ^ 2 + 2x-2 = 0
When X1 = 0.6285 and Y1 = 0.4582, the minimum value of Z is 3.8456
When x2 = -0.6838 and y2 = -0.2548, the maximum value of Z is 15.9148