Question

Finding a Higher-Order Derivative In Exercises \(99-102\) find the given higher- order derivative. $$ f^{\prime}(x)=x^{2}, \quad f^{\prime \prime}(x) $$

Short answer

The second derivative of the function is \(f''(x) = 2x\).

Step-by-step solution

Identify the First Derivative Given

The first derivative is given by \(f'(x) = x^{2}\). This is explicitly provided in the exercise.

Differentiate to Find the Second Derivative

To find the second derivative \(f''(x)\), apply the power rule of differentiation. The power rule states that the derivative of \(x^{n}\) is \(n*x^{n-1}\). Applying this rule on the first derivative \(f'(x) = x^{2}\), we get \(f''(x) = 2x^{2-1}\). By simplifying \([2-1]\) to 1, we get \(f''(x) = 2x^{1}\). This step finally simplifies to \(f''(x) = 2x\).

State the Second Derivative

The second derivative of the function is \(f''(x) = 2x\). This is the final solution for the exercise.

Key concepts

First Derivative

The first derivative of a function provides us with important information about the behavior of the function. It tells us how the function is changing at any given point. In mathematical terms, the first derivative, denoted as \( f'(x) \), represents the slope or the rate of change of the function \( f(x) \). For polynomial functions like \( x^2 \), calculating the first derivative involves applying basic differentiation techniques. In this exercise, the first derivative is given as \( f'(x) = x^2 \). This is a simple polynomial expression, and finding higher-order derivatives from this involves straightforward application of the differentiation rules.

Second Derivative

The second derivative gives us further insight into the function's behavior. It tells us about the concavity of the function or the "curvature" of its graph. With the second derivative \( f''(x) \), we can determine whether the function is concave up or concave down at specific intervals. For this exercise, we used the first derivative \( f'(x) = x^2 \) and differentiated it once more to find the second derivative: \( f''(x) = 2x \). This shows us that the function's curvature increases linearly with \( x \). Thus, the second derivative helps in understanding the acceleration of the rate of change of the function.

Power Rule

The power rule is one of the most common and simplest rules in calculus for differentiation. It states that if you have a function \( x^n \), its derivative is \( n \times x^{n-1} \). This rule makes finding derivatives of polynomial functions much easier.In our exercise, we applied the power rule to the first derivative \( f'(x) = x^2 \). By bringing down the exponent as a coefficient and subtracting one from the exponent, we found the second derivative, resulting in \( f''(x) = 2x \).Tips for Applying the Power Rule:

• Identify terms with powers of \( x \).
• Apply the rule to each term separately.
• Simplify the expression for clarity.

Differentiation Technique

Differentiation is a key concept in calculus, used to find the derivative of a function. This process involves breaking down functions into simpler components and applying rules like the power rule, as seen in this exercise.To differentiate polynomial functions, such as \( f(x) = x^2 \), one usually employs straightforward techniques like:
- The Power Rule- The Sum Rule (derivative of a sum is the sum of derivatives)By understanding these basic differentiation techniques, students can tackle more complex problems with confidence.Ultimately, employing the correct differentiation techniques is crucial for solving higher-order derivatives, which provide deeper insights into a function's behavior at various levels.