Final answer:
To find dy/dx for the function y=2.1 cos(cos(x)) + tan(ysinh(x)), we use the chain rule, product rule, and implicit differentiation. The correct derivative involves a combination of these techniques applied to both trigonometric and hyperbolic functions.
Step-by-step explanation:
The student has asked for the derivative of the function y=2.1 \cos(\cos(x)) + \tan(y\sinh(x)) with respect to x. This problem requires the use of chain rule, product rule, and implicit differentiation techniques due to the complexity of the function and the trigonometric and hyperbolic functions involved.
First, we find the derivative of the first term which is straightforward: d/dx of 2.1 \cos(\cos(x)) yields -2.1 \sin(\cos(x))(-\sin(x)), using the chain rule. The second term involves implicit differentiation since y itself is a function of x. We get d/dx of \tan(y\sinh(x)) using the product and chain rule, and this yields \sec^2(y\sinh(x))(y\cosh(x) + y'\sinh(x)), where y' is the derivative of y with respect to x.
To solve for y', we set these equal to dy/dx and solve, knowing that dy/dx appears in the second term. After manipulation of the terms, we find that the required derivative should match one of the options given in the multiple-choice list, taking into account trigonometric identities and properties of hyperbolic functions.