Differentiate xⁿ, for rational values of n, and related constant…

The idea

This section builds the core library of derivatives you apply throughout calculus. For a power xⁿ with rational n, the derivative is n x^(n−1); this works for positive, negative and fractional indices, so d/dx(x^(1/2)) = (1/2)x^(−1/2) and d/dx(x^(−2)) = −2x^(−3). Constant multiples, sums and differences differentiate term by term: if y = 5x³ − 2x^(1/2) + 7, then dy/dx = 15x² − x^(−1/2). For exponentials, d/dx(e^(kx)) = k e^(kx), and d/dx(a^(kx)) = k a^(kx) ln a, so d/dx(2^(3x)) = 3·2^(3x) ln 2. For trigonometric functions with argument kx, d/dx(sin kx) = k cos kx, d/dx(cos kx) = −k sin kx and d/dx(tan kx) = k sec² kx. The natural logarithm has d/dx(ln x) = 1/x for x > 0. Combine these rules with constant multiples and sums, for example d/dx(4 sin 3x − 2e^(5x)) = 12 cos 3x − 10e^(5x).

What earns the marks

• Applies the power rule n x^(n−1) correctly to rational indices, including negative and fractional powers such as x^(−3) and x^(2/3).
• Differentiates constant multiples, sums and differences term by term, preserving coefficients and signs.
• States and uses d/dx(e^(kx)) = k e^(kx) and d/dx(a^(kx)) = k a^(kx) ln a with the correct constant k and base factor.
• Differentiates sin kx, cos kx and tan kx as k cos kx, −k sin kx and k sec² kx respectively.
• Uses d/dx(ln x) = 1/x, including within sums and constant multiples such as 7 ln x.

Common mistakes and how to put them right

• Forgetting to multiply by the inner constant k when differentiating e^(kx), sin kx or cos kx.

  Put it right by writing the factor k explicitly before the function.

• Subtracting one from the coefficient instead of the index when differentiating a power.

  Put it right by reducing only the exponent, as in x⁵ becoming 5x⁴.

• Treating ln x as differentiating to ln x or to x^(−1) written incorrectly.

  Put it right by writing the derivative as 1/x, valid for x > 0.

Exam tips

• Rewrite roots and reciprocals as fractional or negative indices before differentiating, for example √x as x^(1/2).
• Check each term separately, then combine, so a sign error in one term does not spread through the whole expression.
• When a base is not e, such as 5^(2x), include ln 5 as a factor rather than leaving the derivative as a multiple of the original function alone.