Question

If g(x) = sin(x), find g'(x) and g"(x).

Answer 1

The first derivative of g(x) = sin(x) is g'(x) = cos(x), and the second derivative is g"(x) = -sin(x), found by standard calculus differentiation rules.

Step-by-step explanation:

To find the first and second derivatives of g(x) = sin(x), we can apply standard calculus rules. The first derivative of the sine function, denoted as g'(x), is the cosine function. Hence, g'(x) = cos(x). Applying differentiation a second time to find the second derivative, g"(x), results in differentiating cos(x), which gives us g"(x) = -sin(x).

The sequence of differentiation steps is:

1. g(x) = sin(x)
2. g'(x) = d/dx [sin(x)] = cos(x)
3. g"(x) = d/dx [cos(x)] = -sin(x)

Thus, g'(x) = cos(x) and g"(x) = -sin(x).