Explain in great detail how the derivative rules for polynomials are derived. Afterwards, give some example questions on the derivative rule for polynomials.

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The derivative is fundamentally defined as a limit. We start with the formal definition: f prime of x equals the limit as h approaches zero of f of x plus h minus f of x, all divided by h. This expression represents the slope of a secant line that becomes a tangent line as h approaches zero, giving us the instantaneous rate of change at any point. Now let's derive the power rule from the limit definition. Starting with f of x equals x to the n, we apply the definition of derivative. Using the binomial theorem, we expand x plus h to the n power. After expanding, we can factor out h from the numerator and cancel it with the denominator. As h approaches zero, all terms containing h vanish, leaving us with n times x to the n minus 1. This gives us the power rule: the derivative of x to the n is n times x to the n minus 1. Let's derive the rules for constants and linear functions. For a constant function f of x equals c, applying the limit definition gives us c minus c over h, which equals zero over h. Since this is always zero, the derivative of any constant is zero. For a linear function f of x equals c times x, we get c times x plus h minus c times x over h, which simplifies to c times h over h, equaling c. Therefore, the derivative of c times x is simply c. These rules show that constants have zero slope, while linear functions have constant slope. The sum rule states that the derivative of a sum equals the sum of derivatives. To prove this, we start with the limit definition applied to f plus g. We can separate the numerator into individual function terms, then split the fraction into two separate limits. Using the limit properties, we can separate this into two individual limits, which are simply the derivatives of f and g. This visual shows how the slopes of individual functions add together to give the slope of their sum. Now we can combine all our rules to differentiate any polynomial. For a general polynomial with terms from x to the n down to constants, we apply the power rule to each term, the constant multiple rule, and the sum rule. Let's work through an example: f of x equals 3x to the fourth minus 2x cubed plus 5x squared minus 7x plus 1. We differentiate each term separately: 3x to the fourth becomes 12x cubed, negative 2x cubed becomes negative 6x squared, 5x squared becomes 10x, negative 7x becomes negative 7, and the constant 1 becomes 0. Combining these gives us f prime of x equals 12x cubed minus 6x squared plus 10x minus 7.