Derivative Calculations

In Exercises 1–12, find the first and second derivatives.

y = x³/3 + x²/2 + x/4

검증된 단계별 안내

Step 1: Identify the function for which you need to find the derivatives. The given function is \( y = \frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{4} \).

Step 2: To find the first derivative, apply the power rule to each term of the function. The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \).

Step 3: Differentiate each term separately: \( \frac{d}{dx} \left( \frac{x^3}{3} \right) = x^2 \), \( \frac{d}{dx} \left( \frac{x^2}{2} \right) = x \), and \( \frac{d}{dx} \left( \frac{x}{4} \right) = \frac{1}{4} \). Combine these results to get the first derivative.

Step 4: To find the second derivative, differentiate the first derivative. Apply the power rule again to each term of the first derivative.

Step 5: Differentiate each term of the first derivative: \( \frac{d}{dx} (x^2) = 2x \), \( \frac{d}{dx} (x) = 1 \), and \( \frac{d}{dx} \left( \frac{1}{4} \right) = 0 \). Combine these results to get the second derivative.

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주요 개념

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Power Rule for Derivatives

The power rule is a basic principle in calculus used to find the derivative of a function of the form f(x) = x^n. The rule states that the derivative, f'(x), is n*x^(n-1). This rule simplifies the process of differentiation, allowing us to easily find the rate of change of polynomial functions.

Sum Rule for Derivatives

The sum rule for derivatives states that the derivative of a sum of functions is the sum of their derivatives. If you have a function y = f(x) + g(x), the derivative y' is f'(x) + g'(x). This rule allows us to differentiate each term in a polynomial separately and then combine the results.

Second Derivative

The second derivative of a function is the derivative of the first derivative, providing information about the curvature or concavity of the original function. It is denoted as f''(x) or d²y/dx². Calculating the second derivative helps in understanding the acceleration or the rate of change of the rate of change of the function.

Derivative CalculationsIn Exercises 1–12, find the first and second derivatives.y = x³/3 + x²/2 + x/4