To answer your questions, let's start with the definition of a derivative.
The derivative of a function represents the rate at which the function's output value is changing with respect to its input value. In other words, it tells us how sensitive the function is to small changes in its input. Mathematically, the derivative of a function f(x) is denoted as f'(x) or dy/dx.
Now let's move on to what the derivative of a function tells us in English. The derivative provides us with information about the slope or the rate of change of the function at any given point. It tells us how steep the graph of the function is at a particular point. If the derivative is positive, it means that the function is increasing, while a negative derivative indicates a decreasing function. A derivative of zero suggests a relatively flat portion of the graph.
Now, let's cover the power rule, quotient rule, and chain rule with examples.
1. Power Rule:
The power rule is used to find the derivative of functions in the form of f(x) = x^n, where n is a constant. The power rule states that we can take the exponent n, multiply it by the coefficient, and reduce the power by 1. For example, if we have f(x) = 3x^2, applying the power rule gives f'(x) = 2 * 3x^(2-1) = 6x.
2. Quotient Rule:
The quotient rule is employed when finding the derivative of functions that are ratios of two functions. It states that to find the derivative of a quotient u(x)/v(x), we take the derivative of u(x), multiply it by v(x), subtract the derivative of v(x) multiplied by u(x), and then divide the entire expression by the square of v(x). Let's assume we have f(x) = (2x+1)/(3x-2). By applying the quotient rule, we get f'(x) = ((3x-2)(2) - (2x+1)(3))/((3x-2)^2).
3. Chain Rule:
The chain rule is used when we have a composition of functions, meaning the output of one function becomes the input of another. Let's consider the function f(x) = (2x^2 + 3x)^4. To apply the chain rule, we differentiate the outer function first and multiply it by the derivative of the inner function. In this example, f'(x) = 4(2x^2 + 3x)^3 * (4x + 3).
Moving on to your next set of questions:
The first derivative provides information about the slope or rate of change of a function. It tells us if the function is increasing or decreasing, the location of any maximum or minimum points (critical points), and where the function is concave up or concave down.
The second derivative gives us details about the concavity or the curvature of the function. It tells us whether the graph of the function is bending upwards (concave up) or downwards (concave down) at a given point. It also helps identify inflection points, which are points where the curvature changes from concave up to concave down or vice versa.
I hope that helps!