If 𝑧=𝑠𝑖𝑛⁡(3𝑥+2𝑦), find its derivative with respect to x

The derivative of 𝑧=𝑠𝑖𝑛⁡(3𝑥+2𝑦) with respect to x is found using the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 1: Identify the outer and inner functions. Here, the outer function is sin(u) and the inner function is u=3x+2y.

Step 2: Find the derivative of the outer function. The derivative of sin(u) with respect to u is cos(u).

Step 3: Find the derivative of the inner function with respect to x. The derivative of 3x+2y with respect to x is 3, because the derivative of a constant times x is just the constant, and y is treated as a constant when taking the derivative with respect to x.

Step 4: Multiply the results from steps 2 and 3 according to the chain rule. So, the derivative of 𝑧=𝑠𝑖𝑛⁡(3𝑥+2𝑦) with respect to x is 3cos(3x+2y).