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Partial Differentiation

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Partial derivative of 1/sqrt(x^2+y^2+z^2)

Partial derivative of 1/sqrt(x^2+y^2+z^2)- (delw)/(delx) = x/sqrt(x^2 y^2 z^2) (delw)/(dely) = y/sqrt(x^2 y^2 z^2) (delw)/(delz) = z/sqrt(x^2 y^2 z^2) Since you're dealing with a multivariable function, you must treat x, y, and z as independent variables and calculate the partial derivative of w, your dependent variable, with respect to x, y, and z When you differentiate with respect to x, you treat y and z There's a factor of 2 missing in all your second derivatives The result is exactly as you'd expect The variable you're differentiating with respect to, matters If it's x, then y is treated as a constant, and vice versa So if the "active" variable is leading in the numerator in one derivative, the same should apply in the other

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Definition of Partial Derivatives Let f(x,y) be a function with two variables If we keep y constant and differentiate f (assuming f is differentiable) with respect to the variable x, using the rules and formulas of differentiation, we obtain what is called the partial derivative of f with respect to x which is denoted by Similarly If we keep x constant and differentiate f (assuming f isThe partial derivative of a function of two or more variables with respect to one of its variables is the ordinary derivative of the function with respect to that variable, considering the other variables as constants Example The partial derivative of 3x 2 y 2y 2 with respect to x is 6xy Its partial derivative with respect to y is 3x 2 4y Find the first partial derivatives of 1 f(x,y) = x^y 2 u = x^(y/z) Homework Equations The Attempt at a Solution f_x = y*x^(y1) f_y = lnx?

Therefore at (1,2,4), we get wx = −4, wy = 4, so that the tangent plane is w = 4−4(x −1)4(y −2), or w = −4x 4y x x y 2B2 a) zx = = ;We will need two quantities to classify the critical points of f(x,y) 1 f xx, the second partial derivative of f with respect to x 2 H = f xxf yy −f2 xy the Hessian If the Hessian is zero, then the critical point is degenerate If the Hessian is nonzero, then the critical point is nondegenerate and we can classify the points in the following manner case(i) If H > 0 and f xx < 0 thenPartial derivative of x/ (x^2y^2) \square!

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By symmetry (interchanging x and y), zy = ;Visit http//ilectureonlinecom for more math and science lectures!In this video I will find the 1st and 2nd partial derivative withrespectto x and y of f(

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