We're asked to find the partial derivatives.
Notice for these two examples,
we have functions of three variables.
To determine the partial of f with respect to x,
we consider y and z constants
and differentiate with respect to x.
To determine the partial of f with respect to y,
we consider x and z constants
and differentiate with respect to y.
And to find the partial of f with respect to z,
we consider x and y constants
and different with respect to z.
For the first example,
we are given f of x comma y comma z equals 5xyz.
To find the partial of f with respect to x,
we treat y and z as constants
and differentiate with respect to x.
The derivative of 5xyz with respect to x
is 5 times the derivative of x with respect to x,
which is 1.
And then we still have times yz
because we are treating y and z as constants.
The partial of f with respect to x is 5yz.
To find the partial of f with respect to y,
we now treat x and z as constants
and differentiate with respect to y.
The derivative of 5xyz with respect to y
is 5x times the derivative of y with respect to y,
which is 1.
And then we still have times a z
because we are treating x and z as constants.
The partial of f with respect to y is 5xz.
To find the partial of f with respect to z,
we now treat x and y as constants
and differentiate with respect to z.
And therefore the partial of f with respect to z
is equal to 5xy
times the derivative of z with respect to z,
which is 1, giving us a partial of f
with respect to z as 5xy.
For our second example,
f of x comma y comma z is equal
to 3 times x to the power of 3.3
times y to the power of 6.2 plus z to the seventh.
To find the partial of f with respect to x,
we treat y and z as constants
and differentiate with respect to x.
The derivative of 3x to the power of 3.3
y to the power of 6.2 with respect to x
is 3 times the exponent of 3.3
times x to the power of 3.3 minus 1, which is 2.3,
because we're treating y as a constant
and we still have times a y to the power of 6.2.
And then we have plus the derivative
of z to the seventh with respect to x.
Because we're treating z as a constant,
the derivative with respect to x is 0.
Simplifying, 3 times 3.3 is 9.9,
giving us 9.9x to the power of 2.3
y to the power of 6.2.
To find the partial of f with respect to y,
we now treat x and z as constants
and differentiate with respect to y.
The derivative of 3x to the power of 3.3
y to the power of 6.2 with respect to y
is 3x to the power of 3.3
times the exponent of 6.2
y to the power of 6.2 minus 1, which is 5.2,
plus the derivative of z to the seventh respect to y,
which is 0, because we're treating z as a constant.
Simplifying, 3 times a 6.2 is 18.6,
giving us 18.6x to the power of 3.3
y to the power of 5.2.
And then to find the partial of f with respect to z,
we treat x and y as constants
and differentiate with respect to z.
Because we're treating x and y as constants,
the derivative of 3x to the power of 3.3
y to the power of 6.2 with respect to z is 0.
And then we have plus the derivative
of z to the seventh with respect to z,
which is 7z to the sixth.
I hope you found this helpful.