Final answer:
Derivatives in mathematics involve rates of change and finding derivatives often involve applying differentiation rules to each component of a function or using partial derivatives. For velocity and acceleration in physics, derivatives indicate the rate at which velocity changes over time. Similarly, the slope of a wave or the behavior of a function can be evaluated using derivatives.
Step-by-step explanation:
Finding the derivative of a function is a fundamental task in calculus, which is an area of mathematics that deals with rates of change and the accumulation of quantities. When asked to find the derivative of a velocity function to obtain acceleration, we apply the rules of differentiation to each of the components of the function individually. If the velocity is a vector function, such as v(t) = (10 - 2t)i + 5j + 5k meters per second, taking the derivative with respect to time gives the acceleration a(t). In this case, acceleration is the vector a(t) = -2i meters per second squared, since the other components are constants and their derivatives are zero.
In problems involving motion, such as those that require finding the slope of a wave at a particular point, we also use partial derivatives. For example, the partial derivative of a function like A sin (kx - wt + φ) with respect to position x, while time t is held constant, would give us the slope of the wave, Ak cos(kx - wt + φ).
When dealing with functions like f(x), it's important to examine the behavior of the function and its derivative. In the case of a function f(x) with a positive value and a positive, but decreasing slope at x = 3, among the options provided, y = x² would be a likely candidate as its slope is positive and decreases with increasing x.