Question

Use the chain rule to find dz/ds and dz/dt, where
z= eˣʸ tan y, x= 5s+3t, y= 2s/3t

Answer 1

The derivatives dz/ds and dz/dt are found by applying the chain rule to the function z = e^{xy} tan(y), where x and y are functions of s and t, respectively. Partial derivatives of z with respect to x and y are computed, followed by derivatives of x and y with respect to s and t. These derivatives are then multiplied together according to the chain rule.

Step-by-step explanation:

To find dz/ds and dz/dt for the function z = exy tan(y), where x = 5s + 3t and y = 2s/3t, we apply the chain rule of calculus. The chain rule is a formula for computing the derivative of the composition of two or more functions.

To find dz/ds, we differentiate z with respect to s while treating t as a constant. Similarly, to find dz/dt, we differentiate z with respect to t while treating s as a constant. Notably, we must also use the product rule when differentiating the function exy.

For example, using the chain rule:

• Compute the partial derivatives of z with respect to x and y.
• Then compute the derivatives of x and y with respect to s and t.
• Finally, multiply the respective derivatives together to find dz/ds and dz/dt.