Question

If \(\mathrm{y}=\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{x})))\) and \(\mathrm{f}(0)=0, \mathrm{f}^{\prime}(0)=1\) then \(\mid[\mathrm{dy} / \mathrm{d} \mathrm{x}]_{\mathrm{x} 0}=\) (a) 0 (b) 1 (c) - 1 (d) 2

Short answer

The derivative of y with respect to x, evaluated at x=0, is 1. Therefore, the correct answer is (b) 1.

Step-by-step solution

Write down the composite function for y

We are given that y = f(f(f(x))). This is a composite function for y in terms of f(x).

Differentiate y with respect to x using the chain rule

Recall the chain rule: dy/dx = dy/du * du/dx, where u is a function of x. In our case, let's find dy/dx by breaking down the composite function: Let u = f(x), v = f(u) = f(f(x)), and y = f(v) = f(f(f(x))). Now we'll differentiate each function with respect to its variable. 1. dy/dv = f'(v) 2. dv/du = f'(u) 3. du/dx = f'(x) Using the chain rule, multiply these derivatives together to get dy/dx: dy/dx = dy/dv * dv/du * du/dx = f'(v) * f'(u) * f'(x)

Evaluate dy/dx at x=0

Since f(0) = 0 and f'(0) = 1, we can evaluate dy/dx at x=0: f'(x) = f'(0) = 1 u = f(x) = f(0) = 0 -> f'(u) = f'(0) = 1 v = f(u) = f(0) = 0 -> f'(v) = f'(0) = 1 At x=0, substitute these values into dy/dx: dy/dx = f'(v) * f'(u) * f'(x) = 1 * 1 * 1 = 1

Choose the correct answer from the given options

Since dy/dx evaluated at x=0 is 1, the correct answer is (b) 1.

Key concepts

Composite Functions

Understanding composite functions is fundamental when dealing with complex mathematical relationships. A composite function is essentially a function within another function, expressed as \( f(g(x)) \).

Consider a scenario where you have two functions, \( f(x) \) and \( g(x) \), creating a composite function \( h(x) = f(g(x)) \). In our exercise, the composite function is even more intricate, involving three layers of the same function: \( y = f(f(f(x))) \).

To grasp the concept better, imagine that each function is a machine that processes input to give an output. The output of the first machine becomes the input for the second, and so on. This chaining of functions is what leads us into the realm of the chain rule when it comes to differentiation.

Derivatives

Derivatives are a cornerstone of calculus and they provide a way to describe the rate at which a function is changing at any given point. Formally, the derivative of a function \( f(x) \) at a point \( x \) represents the slope of the tangent line to the function's graph at that point.

In layman's terms, imagine you're driving and your speedometer is showing your rate of change of position with respect to time—that's a derivative in the real world. The fundamental notation for derivatives is either \( f'(x) \) or \( \frac{df}{dx} \), both of which tell us how the function \( f(x) \) changes as \( x \) changes.

Chain Rule Application

The chain rule is a technique for taking the derivative of a composite function. As in our exercise, when differentiating a composite function like \( y = f(f(f(x))) \) at a specific value of \( x \)—say, \( x=0 \)—it’s like unravelling a nested set of boxes to find the derivative inside.

In practical application, the chain rule states that the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Symbolically, if we have \( y = f(g(x)) \) then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

Applying the chain rule can sometimes feel like operating a conveyor belt in a factory: each derivative you find is a product that moves down the line to be multiplied with the next, until you find the final rate of change.

Function Differentiation

Function differentiation is the process of finding the derivative of a function. It's like discovering the instantaneous 'behavior' of the function at any point. What's crucial in the differentiation of functions, especially composite ones, is to recognize the type of function and apply the appropriate rule.

In an exercise like ours, we are given conditions such as \( f(0) = 0 \) and \( f'(0) = 1 \), which guide us in evaluating the derivative after we've applied the chain rule. We symbolically differentiate layer by layer, and then plug in these numbers to get our solution. This step-by-step approach not only ensures that we keep track of our operations, but also aids in identifying any patterns or simplifications early on in the differentiation process.