Find the derivatives of the functions in Exercises 1–42.


𝔂 = x⁵ - 0.125x² + 0.25x

Verified step by step guidance

1. Identify the function for which you need to find the derivative: 𝔂 = x⁵ - 0.125x² + 0.25x.
2. Apply the power rule for differentiation, which states that the derivative of xⁿ is n*xⁿ⁻¹. Start with the first term, x⁵. The derivative is 5*x⁴.
3. Move to the second term, -0.125x². Using the power rule, the derivative is -0.125 * 2 * x¹, which simplifies to -0.25x.
4. Differentiate the third term, 0.25x. The derivative of x is 1, so the derivative of 0.25x is 0.25.
5. Combine the derivatives of each term to find the derivative of the entire function: 𝔂' = 5x⁴ - 0.25x + 0.25.

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 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 differentiation process for terms involving powers of x.

Sum Rule

The sum rule in calculus states that the derivative of a sum of functions is equal to the sum of their derivatives. If f(x) = g(x) + h(x), then the derivative f'(x) = g'(x) + h'(x). This rule allows for the differentiation of complex functions by breaking them down into simpler components.