Find the first partial derivatives of the function. z = x sin(xy)?

Differentiating with respect to a single variable while keeping other variables as constants in the expression is known as Partial differentiation.

Answer: The first partial derivatives of the function. z = x sin(xy) are sin(xy) + xy cos(xy) and x²cos(xy).

Let's go through the explanation to understand better.

Explanation:

Let z = x sin(xy)

Differentiate the function z = x sin(xy) w.r.t. x, and treat y as a constant.

d[z]/dx = d[ x sin(xy)]/dx

d[z]/dx = 1⋅sin(xy) + cos(xy)(y)⋅x

d[z]/dx = sin(xy) + xy cos(xy)

Now, Differentiate the function z = x sin(xy) w.r.t. y, and treat x as a constant.

d[z]/dy = d[ x sin(xy)]/dy

⇒ d[z]/dy = x⋅cos(xy)(x)

⇒ d[z]/dy = x²cos(xy)

A Cuemath's Partial Derivative Calculator helps to calculate the value of the partial derivatives.

Thus, the first partial derivatives of the function. z = x sin(xy) are sin(xy) + xy cos(xy) and x²cos(xy).