Question

If sin(2x+5y+z)=0, use implicit differentiation to find the first partial derivatives ∂z/∂x and ∂z/∂y at the point (0,0,0).

A. ∂z/∂x(0,0,0) = ________
​B. ∂z/∂y(0,0,0) = ________

Answer 1

To find ⇾z/⇾x and ⇾z/⇾y using implicit differentiation, differentiate the given equation with respect to x and y and substitute the values x = 0, y = 0, and z = 0 into the resulting expressions.

Step-by-step explanation:

To find ⇾z/⇾x and ⇾z/⇾y using implicit differentiation, we'll differentiate both sides of the equation sin(2x+5y+z) = 0 with respect to x and y. Let's start with the partial derivative ⇾z/⇾x.

Using the chain rule, ⇾z/⇾x = -1 / (∂(sin(2x+5y+z)) / ∂x).

Simplifying further, we get ⇾z/⇾x = -1 / (2cos(2x+5y+z)).

To find ⇾z/⇾x at the point (0,0,0), substitute x = 0, y = 0, and z = 0 into the partial derivative expression. Thus, ⇾z/⇾x(0,0,0) = -1 / (2cos(0+0+0)).

Similarly, to find ⇾z/⇾y, we differentiate with respect to y and substitute y = 0, x = 0, and z = 0 into the resulting expression. Thus, ⇾z/⇾y(0,0,0) = -1 / (2cos(0+0+0)).