Find derivatives of polynomials

Learning Objectives Define a polynomial and its derivative. State and apply the Power Rule to find the derivative of terms in the form x^n. Apply the Constant, Constant Multiple, and Sum/Difference rules in conjunction. Find the derivative of any multi-term polynomial function. Calculate the slope of the tangent line to a polynomial curve at a specific point. Determine the equation of the tangent line to a polynomial at a given x-value. Ever wondered how a video game character can smoothly jump along a curved path? 🎮 The secret lies in calculus and finding the precise slope of the curve at every instant! This tutorial will introduce the fundamental rules for finding the derivative of any polynomial function. Mastering these rules is the first major step in differential cal...

TermDefinitionExample Polynomial FunctionAn expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.f(x) = 5x^4 - 2x^3 + x - 7 DerivativeA function, denoted f'(x), that gives the instantaneous rate of change or the slope of the tangent line to the graph of f(x) at any given point.If f(x) = x^2, its derivative is f'(x) = 2x. This means the slope of the tangent line at x=3 is f'(3) = 2(3) = 6. TermA single part of a polynomial expression, separated by plus or minus signs.In the polynomial 5x^4 - 2x^3 + x - 7, the terms are 5x^4, -2x^3, x, and -7. CoefficientThe numerical factor that is multiplied by the variable in a term.In the term -2x^3, the coefficient...

The Power Rule If f(x) = x^n, then f'(x) = n * x^(n-1) This is the most fundamental rule for differentiating polynomials. To find the derivative of a variable raised to a power, you bring the exponent down as a multiplier and then subtract one from the original exponent. The Constant Multiple Rule If h(x) = c * f(x), then h'(x) = c * f'(x) This rule states that a constant coefficient can be carried through the differentiation process. You simply find the derivative of the function part and then multiply it by the constant. The Sum/Difference Rule If h(x) = f(x) ± g(x), then h'(x) = f'(x) ± g'(x) This rule allows you to differentiate a polynomial term by term. The derivative of a sum of terms is the sum of their individual derivatives....