Differentiating power functions

Task goals

• Differentiate a single power of x, with or without a whole-number coefficient, by multiplying by the exponent and reducing the exponent by one (e.g. f(x) = x^2, f(x) = x^4, f(x) = 2x^2, f(x) = 5x^4).
• Differentiate a polynomial term by term, including linear terms, and recognise that the derivative of a constant term is zero (e.g. f(x) = 6x^2 + 9x - 3, f(x) = 3x^4 + 6x^3 - 7x^2 - 2x + 3, f(x) = 7, f(x) = 4x^5 + 8).
• Apply the power rule to polynomials whose coefficients are fractions or decimals, keeping the coefficient arithmetic exact (e.g. f(x) = (2/3)x^3 + (1/2)x^2 - 5x + 1, f(x) = 3.6x^5 - 1.2x^4 + 2.8x^3 - 4.3x^2 + 6).
• Expand a product of factors into an expanded polynomial first, then differentiate the result (e.g. f(x) = (2x - 1)(x + 3), f(x) = x(x - 3)(x + 2), f(x) = 4x^3(3x^2 - 2x + 1)).
• Rewrite reciprocal terms as negative powers of x before differentiating, and give the derivative of each (e.g. f(x) = 4x^7 + 2x - 3 + 1/x, f(x) = 2/x^3 + 3x^2 - 4x + 1/x, f(x) = 1/x^2 + 2x^3 - 3/x).
• Simplify an algebraic fraction by dividing every term of the numerator by the denominator, then differentiate the resulting powers of x (e.g. f(x) = (4x^3 + 7x^2 - 6x - 5)/x^2, f(x) = (2x^3 + 6x^2 - 4x)/(2x), f(x) = (4x^6 - 8x^4 + 2x^2)/(2x^2)).
• Rewrite a root, or the reciprocal of a root, as a fractional or negative fractional power of x and then differentiate it (e.g. f(x) = sqrt(x), f(x) = cube root of x, f(x) = sqrt(x^3), f(x) = 3/sqrt(x), f(x) = 5/(cube root of x^2)).
• Differentiate multi-term expressions that combine negative and fractional exponents, including quotients containing surds that must be simplified before the power rule is applied (e.g. f(x) = 1/x^3 - 4/x^4 + 1/(cube root of x^2), f(x) = (2x^2 + sqrt(x))/x, f(x) = (2 sqrt(x^3) + 3x - 5 (fourth root of x))/x).