Question

Find the derivative of y=sin(πx)², and I got πi sin(2πx) however, the answer key says 2π²xcos(πx)². How to play around to get the answer on the answer key?

Answer 1

To correctly derive y = sin(πx)², apply the chain rule, differentiating the outer function, which gives cos(u), and then the inner function, which gives 2πx. The final answer is dy/dx = 2π²x cos((πx)²).

Step-by-step explanation:

The derivative of the function y = sin(πx)² involves applying the chain rule. Firstly, let's denote u = (πx)². Then, sin(u) is the outer function and u is the inner function. The derivative of sin(u) with respect to u is cos(u), and the derivative of u with respect to x is 2πx. By the chain rule, the derivative of y with respect to x is:

dy/dx = d(sin(u))/du * du/dx = cos((πx)²) * 2πx

Finally,

dy/dx = 2π²x cos((πx)²)

The confused student might have incorrectly applied the product or chain rule, perhaps attempting to take the derivative of sin as if it were an exponential function, which would incorrectly introduce i into the result.