Question

Найдите вторые производные y = lnsinx

Answer 1

G(x)    =    y "   =  (lnsinx) "  =  1/sinx *(sinx) "  =1/sinx*cosx    =   ctgx  первая  производная

 

G ' (x)  =  (ctgx) "   =    -1/sin^2x      вторая  производная

Answer 2

Finding the Second Derivative of y = ln(sin(x))

To find the second derivative of y = ln(sin(x)), we'll first need to find the first derivative of y with respect to x, and then find the second derivative by differentiating the first derivative with respect to x.

1. First Derivative of y = ln(sin(x)): The first derivative of y = ln(sin(x)) can be found using the chain rule, which states that if y = f(g(x)), then y' = f'(g(x)) * g'(x).

Applying the chain rule: - y = ln(u), where u = sin(x) - y' = (1/u) * u', where u' is the derivative of sin(x)

Using the derivatives of sin(x) and ln(u): - y' = (1/sin(x)) * cos(x) - y' = cot(x) 2. Second Derivative of y = ln(sin(x)): To find the second derivative, we need to differentiate the first derivative y' = cot(x) with respect to x.

Differentiating y' = cot(x): - y'' = -csc^2(x)

Therefore, the second derivative of y = ln(sin(x)) is: - y'' = -csc^2(x) So, the second derivative of y = ln(sin(x)) is y'' = -csc^2(x).

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