Final answer:
The partial derivative of the function f with respect to x, for a sinusoidal wave equation, is found by applying the chain rule, resulting in Ak cos(kx - wt + d).
Step-by-step explanation:
The partial derivative of a function with several variables is the derivative with respect to one of those variables, with the others held constant. In the provided example, the partial derivative of the function f with respect to x is given for f in the form of a sinusoidal wave equation. To find this first derivative with respect to x, the chain rule is applied to the sine function, holding the time variable constant. The derivative of A sin (kx - wt + d) with respect to x is Ak cos(kx - wt + d). The process involves differentiating the inside of the sine function which is linear in x, resulting in a cosine function with the same arguments, multiplied by the constant factor that is the derivative of the linear function inside the sine with respect to x.