Question

Let \(f(0)=1\) and \(f^{\prime}(0)=2\). Find the derivative of \(f(f(x)-1)\) at \(x=0\).

Short answer

The derivative is 4.

Step-by-step solution

Define the function

Let \( g(x) = f(f(x) - 1) \). We need to find \( g'(x) \). By the chain rule, \( g'(x) = f'(f(x) - 1) \cdot (f'(x)) \).

Evaluate at x = 0

Since we want the derivative at \( x = 0 \), we find \( g'(0) \). First, determine \( f(x) - 1 \) at \( x = 0 \). Substitute \( x = 0 \) in \( f(x) - 1 \) to get \( f(0) - 1 = 1 - 1 = 0 \).

Apply composition derivative at x = 0

Next, evaluate \( g(x) = f(f(x) - 1) \) at \( f(x) - 1 = 0 \). Therefore, \( g(x) = f(0) \) when \( f(x) - 1 = 0 \). Finding \( g'(0) \) involves applying the chain rule: \( g'(0) = f'(f(0) - 1) \cdot f'(0) = f'(0) \cdot f'(0) \).

Plug in known values

From the question, \( f(0) = 1 \) and \( f'(0) = 2 \). Therefore, \( g'(0) = 2 \times 2 = 4 \).

Key concepts

Derivative Calculation

Calculating derivatives is one of the foundational skills in calculus. Derivatives measure how a function changes as its input changes. In simpler terms, it provides the rate at which a particular function is changing at any given point.

• Think of a derivative as the slope of a curve at a specific point;
• It tells you if the function is increasing, decreasing, or staying constant.

To find a derivative, we can use different rules and techniques depending on the form of the function. For basic polynomial functions, apply the power rule, which involves bringing down the exponent and reducing it by one.
Derivatives are not just theoretical but have practical applications such as forecasting in economics and natural sciences. Overall, derivatives are crucial for understanding relationships within mathematical models.

Chain Rule

The chain rule is an essential technique for calculating the derivative of composite functions. Composite functions are those where one function is nested inside another. Essentially, the chain rule allows you to differentiate these types of functions by differentiating each component separately and then combining the results.
To apply the chain rule, follow three steps:

• Differentiating the outer function first;
• Then multiply by the derivative of the inner function;
• Keep the "chain" intact by focusing on each part.

This method is particularly handy when dealing with more complex functions like trigonometric, exponential, or logarithmic forms. For instance, if you have a function like \( h(x) = \sin(3x) \), you would first differentiate \( \sin(u) \) with respect to \( u \) and then multiply by \( 3 \) (the derivative of \( 3x \)). This way, the chain rule makes dealing with nested functions straightforward.

Function Composition

Function composition is all about stacking functions, meaning taking the output of one function and using it as the input for another. It’s like putting on socks first, then shoes—the socks are crucial for the shoes to be worn correctly. In notation terms, for two functions \( f(x) \) and \( g(x) \), the composition is written as \( f(g(x)) \).
Function composition is a powerful way to create more complex functions from simpler ones. It can also help model real-world scenarios where several steps or transformations are applied sequentially.

• Composing functions can also simplify problems;
• By breaking them down into smaller, more manageable parts.

Understanding function composition is also beneficial when deriving functions using the chain rule, as it provides the structural understanding needed for those calculations.

Evaluating Derivatives at a Point

Evaluating the derivative of a function at a specific point reveals the behavior of the function exactly at that point. It's like zooming in to understand precisely how fast the function is moving at just that instance. This is particularly useful in finding tangent lines or instantaneous rates of change.
To evaluate a derivative at a point, follow these steps:

• Calculate the general derivative;
• Plug in the specific x-value you are interested in.

For the given exercise, this involved finding \( g'(0) \) for the function \( g(x) = f(f(x) - 1) \). By applying the chain rule, we could determine the derivative in a simplified form and then plug in our known values.
This meticulous process gives us insights into the precise change rate, which is infinitely valuable in analytical scenarios.