Question

Y=x^2+1/sinh(x) compute the first derivative

a)2x-coth(x)
b)2x+cosh(x)
c)2x- csch(x)
d)2x+ sech(x)

Answer 1

The first derivative of the function Y = x^2 + 1/sinh(x) is calculated using the power rule for x^2, which yields 2x, and the chain rule for 1/sinh(x), yielding -csch(x). The combined result for the first derivative is 2x - csch(x).

Step-by-step explanation:

The student is asking about how to find the first derivative of the function Y = x^2 + 1/sinh(x).

To compute the first derivative, we need to apply the rules of differentiation, which include the power rule, the constant factor rule, and the chain rule. Additionally, we need knowledge of the derivatives of hyperbolic functions.

Step-by-step, the differentiation process for this function is as follows:

1. Differentiate x^2, which by power rule gives us 2x.
2. Differentiate 1/sinh(x), which is the same as sinh(x)^(-1). By the chain rule and knowing the derivative of csch(x) is -coth(x)csch(x), this part gives us -csch(x)

Therefore, combining these two parts we get the first derivative, dY/dx = 2x - csch(x).