How to solve a Gradient Vector question with a square root?

What is the gradient vector of the function





f(x,y,z)= 9 SQRT(x^2+y^2+z^2)





Ive tried to solve it several different ways and i still can figure it out, i think im not converting the SQRT to exponents correctly.|||grad F = 鈭侳/鈭倄I + 鈭侳/鈭倅J + 鈭侳/鈭倆K





I,J,K are usual unit vecdtors.








鈭侳/鈭倄 = 鈭?鈭倄[9鈭?x虏 + y虏 + z虏)]





= 9[1/2 (x虏 + y虏 + z虏)^-1/2 X(2x)]





鈭侳/鈭倄 = 9x / 鈭?x虏 + y虏 + z虏)





By symmetry





鈭侳/鈭倅 = 9y / 鈭?x虏 + y虏 + z虏)








鈭侳/鈭倆 = 9z / 鈭?x虏 + y虏 + z虏)








gradF = [9x /鈭?x虏+ y虏+z虏)]I + [9y /鈭?x虏+y虏+z虏)]J + [9z /鈭?x虏+y虏+z虏)]K





= [9 / 鈭?x虏+y虏+z虏)] [xI + yJ + zK]





Remember vector %26lt; r %26gt; = xI + yJ + zK





also r虏 = x虏+y虏+z虏





so r = 鈭?x虏+y虏+z虏)





Hence gradF = 9(%26lt; r %26gt; / r)





Note: there is no easy way to represent vectors here so I have





used %26lt; r %26gt; to represent actual vector and r to represent magnitude of vector.