The general polynomial of degree n has the form
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀,
where aₙ ≠ 0. Find P'(x).

Verified step by step guidance

1. To find the derivative of the polynomial P(x), apply the power rule for differentiation, which states that the derivative of xⁿ is n*xⁿ⁻¹.
2. Start with the first term aₙxⁿ. The derivative is aₙ * n * xⁿ⁻¹.
3. Proceed to the next term aₙ₋₁xⁿ⁻¹. The derivative is aₙ₋₁ * (n-1) * xⁿ⁻².
4. Continue applying the power rule to each term: a₂x² becomes 2*a₂*x, and a₁x becomes a₁.
5. The constant term a₀ has a derivative of 0, as constants disappear when differentiated. Combine all these derivatives to form P'(x).

Key Concepts

Polynomial Function

A polynomial function is an expression consisting of variables and coefficients, structured in terms of powers of a variable. The general form is P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where n is a non-negative integer, and aₙ ≠ 0. Understanding the structure of polynomials is crucial for applying differentiation rules.

Power Rule for Differentiation

The power rule is a basic principle in calculus used to differentiate functions of the form xⁿ. It states that the derivative of xⁿ is nxⁿ⁻¹. This rule is essential for finding the derivative of each term in a polynomial, allowing us to compute the derivative of the entire polynomial function.

Derivative of a Polynomial

The derivative of a polynomial function is obtained by applying the power rule to each term individually. For P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, the derivative P'(x) is aₙnxⁿ⁻¹ + aₙ₋₁(n-1)xⁿ⁻² + ... + a₁. This process involves reducing the power of each term by one and multiplying by the original power.