Question

Explain in your own words how to find the third derivative of a function \(f(x)\).

Short answer

Take three derivatives of \\(f(x)\\) to find \\(f'''(x)\\).

Step-by-step solution

Understanding Derivatives

The derivative of a function, denoted as \(f'(x)\), represents the rate at which the function changes with respect to \x\. To find higher-order derivatives, we keep differentiating the function multiple times. In our task, we need the third derivative, denoted as \(f'''(x)\).

Find the First Derivative

Start by finding the first derivative of \(f(x)\). Use the rules of differentiation such as the power rule, product rule, or chain rule as needed. For instance, if \(f(x) = x^n\), the first derivative is \(f'(x) = nx^{n-1}\).

Calculate the Second Derivative

Differentiate the first derivative \(f'(x)\) to get the second derivative \(f''(x)\). Again, apply differentiation rules to find this. Continuing our example, for \(f'(x) = nx^{n-1}\), the second derivative is \(f''(x) = n(n-1)x^{n-2}\).

Compute the Third Derivative

Derive the second derivative \(f''(x)\) to find the third derivative \(f'''(x)\). Use appropriate differentiation rules once more. In our ongoing example, the third derivative would be \(f'''(x) = n(n-1)(n-2)x^{n-3}\).

Verify the Results

Double-check each differentiation step to ensure accuracy. Confirm that you have applied the correct differentiation rules at each step and that calculations are correct.

Key concepts

Differentiation Rules

Differentiation is a key process in calculus. It involves finding the derivative of a function. A derivative is simply a measure of how a function changes as its input changes, which is vital for understanding dynamic systems.

To find a derivative, we use specific rules. These differentiation rules include:

• Power Rule: This rule states that if you have a function of the form \(f(x) = x^n\), the derivative is \(f'(x) = nx^{n-1}\).
• Product Rule: Whenever you have a function that is the product of two other functions, say \(g(x)\) and \(h(x)\), the derivative is given as \((g \cdot h)' = g' \cdot h + g \cdot h'\).
• Chain Rule: When you have a function composed of another function, such as \(f(g(x))\), the chain rule helps differentiate it by finding \(f'(g(x)) \cdot g'(x)\).

These rules simplify the process of differentiation, making it manageable even for complex functions.

Identifying which rule to apply is crucial for calculating derivatives accurately.

Higher-Order Derivatives

Once you master the basic derivative, the next step is exploring higher-order derivatives. A higher-order derivative is the derivative of a derivative. This process involves differentiating a function multiple times.

For example, the first derivative \(f'(x)\) gives the initial rate of change of the function \(f(x)\).
The second derivative \(f''(x)\) tells us about the curvature or concavity of the original function.

• Second Derivative: Differentiating \(f'(x)\) gives us \(f''(x)\), indicating how the rate of change itself changes.
• Third Derivative: When differentiating again to get \(f'''(x)\), we find even deeper layers of change, sometimes interpreted in physics as the rate of change of acceleration.

Continuing the process will lead to fourth, fifth, and even higher derivatives. Understanding higher-order derivatives gives insight into the more intricate behaviors of functions.

Rate of Change

The concept of a derivative is fundamentally tied to the idea of rate of change.
A derivative measures how a quantity changes with respect to another. This is vital in various fields like physics, economics, and biology.

In a simple example, if we consider the position of an object over time with the function \(s(t)\), the first derivative \(s'(t)\) gives us velocity, indicating how fast the position changes.

• The first derivative represents the immediate rate of change.
• The second derivative indicates the rate at which the rate of change itself is changing. For instance, in our example, it gives us acceleration.
• The third derivative goes even deeper, describing the rate of change of acceleration, sometimes called jerk.

Each derivative provides a new layer of understanding, helping us to see not just how a function changes, but how its changing speed and acceleration evolve over time.