Final answer:
To find the derivative of the function f(x) = 3x tan⁻¹ (x²) e sin x, one must apply the product rule, the chain rule, and the rules for deriving inverse trigonometric functions and exponential functions.
Each part of the function is differentiated in turn, while treating the rest as a constant, and then all the derivatives are combined using the product rule.
Step-by-step explanation:
The student is asking how to find the derivative of the function f(x) = 3x tan⁻¹ (x²) e sin x. This requires the application of several rules of differentiation, including the product rule, the chain rule, and the derivative of inverse trigonometric functions. In our case, the function is the product of three functions, so we have to apply the product rule which states that the derivative of a product of functions is given by:
d/dx [u(x)v(x)w(x)] = u'(x)v(x)w(x) + u(x)v'(x)w(x) + u(x)v(x)w'(x).
Using this rule involves differentiating each part of the function while keeping the others constant one by one. Then, for example, to differentiate tan⁻¹ (x²), we would use the chain rule, since it is a composite function (an inverse trigonometric function of x²). The derivative of tan⁻¹ u with respect to u is 1/(1+u²), and so in this case, the derivative with respect to x would involve multiplying by the derivative of x², which is 2x.
Finally, the derivative of e sin x would use the chain rule, where the outer function is the exponential function and the inner function is sin x. The derivative of eˣ, where u is a function of x, is eˣ • u'. So for e sin x, the derivative would be e sin x • cos x.
Combining all these derivatives using the product rule will yield the derivative of the whole function.