Final Answer:
For h(x) = sin(x), the first derivative, h'(x), is h'(x) = cos(x), and the second derivative, h''(x), is h''(x) = -sin(x).
Step-by-step explanation:
The function h(x) = sin(x) represents the sine function. To find the first derivative h'(x), differentiate h(x) with respect to x. Applying the differentiation rule for the sine function, the derivative of sin(x) is cos(x). Therefore, h'(x) = cos(x), signifying that the rate of change of h(x) (the sine function) with respect to x is given by the cosine function.
To calculate the second derivative h''(x), which represents the rate of change of h'(x) (the cosine function) with respect to x, differentiate h'(x) with respect to x. The derivative of cos(x) is -sin(x), resulting in h''(x) = -sin(x). This signifies that the rate of change of the cosine function h'(x) with respect to x is given by the negative sine function.
Therefore, the first derivative h'(x) of the sine function h(x) = sin(x) is cos(x), and the second derivative h''(x) of the same function is -sin(x). These derivatives describe the instantaneous rates of change of the original function and its first derivative, respectively, with respect to x.