00:06This video presents a series of exercises related to partial derivatives.
00:11Their definition is as follows: they are the derivatives of a function of several variables with respect to
00:15For each of the variables, keeping the others constant, its symbol is the letter
00:20delta, to distinguish it from other derivatives.
00:23Its logic is the partial derivative in this case of the function with respect to x, when
00:27and is constant, and the differentiation is performed, keeping in mind y as a constant.
00:33It is also the partial derivative of the function with respect to y, when x is constant, and
00:38The differentiation is performed, keeping in mind x as a constant.
00:42Thus we have the following exercises, finding the partial derivative of the function of z is
00:46equal to, a x squared, plus, bxy, plus y squared, plus dx, plus,
00:53e y, plus f. This is the partial derivative of z with respect to x, when y is constant,
01:00derived the second member to the first derivative of x squared, where a is a constant,
01:05plus, and constant times the derivative of bx, plus, the derivative of c squared, plus,
01:11d times the derivative of x, plus, y times the derivative of e, plus, the first derivative of f constant.
01:18We get the answer 2ax, plus, by, plus, t.
01:23Now for the partial derivative of z with respect to y, when x is constant, we have, the first
01:29derivative of y, times the derivative of ax squared, plus x times the derivative of by,
01:35plus, the derivative of c squared, plus x times the derivative of d, plus the derivative,
01:40b and, plus the derivative of the constant f. We obtain as an answer bx, plus, 12y, plus, e.
01:49Continuing with the exercises, we have the function xy, equal to x cubed, plus
01:543bx squared y, plus 13xy squared, plus d, y cubed.
02:01We have the function notation xy, we can keep x variable here and y fixed or constant,
02:06This is equal to the derivative of x cubed, plus, 3by the derivative of x raised
02:12squared, plus, 13y squared, by the derivative of x, plus, d y raised to the
02:18cube. We obtain the following answer, factoring, 3 which multiplies x squared, plus
02:242byx, plus ey.
02:26Now, we have the function notation of xy; we can keep y variable here and x fixed or constant.
02:33This is equal to the derivative of x cubed, plus, 3bx squared times
02:38the derivative of y of y, plus, 13x times the derivative of y of y squared, plus, d
02:44by
02:44the derivative of y, of y cubed.
02:47We obtain as an answer, by common factor, 3 which multiplies bx squared, plus
02:5312xy, more than y squared.
02:56The next exercise is to find the partial derivative of the following function, a x, plus
03:01b and, over cx, more than y.
03:04The notation tells us that y is constant or fixed and x is variable; this is equal to the derivative.
03:09from the quotient, to cx, plus y, in parentheses, by the derivative of x, in parentheses to
03:15x, plus b, and minus, in parentheses x, plus b, and multiplying cx, plus y, all
03:23over the square in parentheses of cx, plus, of y.
03:26And, simplifying and multiplying, we obtain, ad, plus bc, multiplied by and over cx, plus
03:33of y, this squared.
03:36The notation tells us that x is constant or fixed and y is variable; this is equal to the derivative.
03:41of the quotient, in parentheses, cx, plus b and, by the first derivative of y, which multiplies
03:47in parentheses a x, plus b y, minus in parentheses a x, plus b y, which multiplies the
03:53derivative of y, times cx, plus y, all this over cx, plus y raised
03:59squared.
04:00Simplifying and reducing, we obtain, in parentheses bc, plus ad, times x, this over
04:06cx, plus y, this squared.
04:09Now we have the limits, the standard limits apply, so in the exercise we
04:14It says, the limit xy implies minus 1, 2, of x, minus 2y, over x squared, plus xy, is
04:22The same as replacing the values in the variables, we have minus 5 thirds.
04:26We have another exercise, the limit of xy, implies 0,0 of x, minus y over x squared, minus y
04:34square, is equal to an indeterminate limit.
04:36The last exercise tells us, the limit xy implies 3,2 is equal to the expression 2xy,
04:42negative 7x squared, negative 7x squared, and squared equals to replace values, equals to negative
04:48156.
04:50That's all. If you found the video helpful, don't forget to subscribe.
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