Final answer:
To differentiate the function y = -5e³x²−csc(7x), we use the chain rule for both the exponential and trigonometric parts, yielding the derivative dy/dx = -30xe³x² - 7cot(7x)csc(7x).
Step-by-step explanation:
The student has asked to differentiate the function y = -5e³x²−csc(7x). To do this, we apply the rules of differentiation. The derivative of the first term, with respect to x, involves the exponential function and the chain rule, and the second term is a trigonometric function for which we also use the chain rule.
The derivative of -5e³x² with respect to x is -30xe³x² because we multiply by the exponent's derivative according to the chain rule.
The derivative of -csc(7x) with respect to x is 7cot(7x)csc(7x), also using the chain rule and trigonometric identities. Combining these results gives us the total derivative of y with respect to x as dy/dx = -30xe³x² - 7cot(7x)csc(7x).