Derivative Calculations


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


y = x² + x + 8

Verified step by step guidance

1. Step 1: Identify the function for which you need to find the derivatives. In this case, the function is y = x² + x + 8.
2. Step 2: To find the first derivative, apply the power rule to each term of the function. The power rule states that if y = xⁿ, then the derivative dy/dx = n*x^(n-1).
3. Step 3: Apply the power rule to the first term x². The derivative of x² is 2*x^(2-1) = 2x.
4. Step 4: Apply the power rule to the second term x. The derivative of x is 1*x^(1-1) = 1.
5. Step 5: The constant term 8 has a derivative of 0, as the derivative of any constant is zero. Combine the derivatives of each term to find the first derivative: dy/dx = 2x + 1. Then, apply the same process to find the second derivative by differentiating dy/dx = 2x + 1 again.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In practical terms, the derivative provides the slope of the tangent line to the curve of the function at any given point.

Power Rule

The Power Rule is a fundamental technique for finding derivatives of polynomial functions. It states that if a function is in the form of f(x) = x^n, where n is a real number, then its derivative f'(x) is given by f'(x) = n*x^(n-1). This rule simplifies the process of differentiation for functions involving powers of x.

Second Derivative

The second derivative of a function is the derivative of the first derivative. It provides information about the curvature of the function's graph and can indicate whether the function is concave up or concave down. The second derivative is useful for analyzing the acceleration of a function and identifying points of inflection where the function changes concavity.