Final answer:
The derivative of the function y = sin(x³ + ln(x) + 4e^x) with respect to x is found by using the chain rule and is y' = cos(x³ + ln(x) + 4e^x) · (3x² + 1/x + 4e^x).
Step-by-step explanation:
To compute y' for the function y = sin(x³ + ln(x) + 4e^x), we need to use the chain rule for differentiation. This rule allows us to find the derivative of composite functions. We'll treat the inside of the sine function, x³ + ln(x) + 4e^x, as a single function that we'll call u(x), then find the derivative of sin(u) with respect to u, and multiply it by the derivative of u with respect to x.
Let u(x) = x³ + ln(x) + 4e^x. The derivative of sin(u) with respect to u is cos(u). Next, we find u' (the derivative of u with respect to x): u' = 3x² + 1/x + 4e^x.
Finally, we use the chain rule to find y': y' = cos(u) · u'. Substituting u back in, we get y' = cos(x³ + ln(x) + 4e^x) · (3x² + 1/x + 4e^x).