Derivatives of standard functions (the formulas)

You already know what the derivative at a point represents: the slope of the tangent line to the curve, that is, how fast the function is changing there. But computing that slope by hand, with a rate of change, is slow. Good news: for the standard functions, the work has been done once and for all, and you just need to know the formulas.

The star of the table is the power rule: the derivative of xⁿ is n·xⁿ⁻¹. The exponent drops down in front as a coefficient, and the exponent goes down by 1. So the derivative of x² is 2x, that of x³ is 3x², that of x⁴ is 4x³. Two extreme cases to remember: the derivative of x is 1, and the derivative of a constant is 0 (a constant function does not change, its slope is zero everywhere).

Two more formulas complete the kit: the derivative of 1/x is -1/x², and the derivative of √x is 1/(2√x). Notice that both are consistent with the power rule, since 1/x = x⁻¹ and √x = x^(1/2).

Finally, two combination rules make the table truly powerful: the derivative of k·f is k·f′ (a multiplicative constant stays as a factor), and the derivative of a sum is the sum of the derivatives. With these, you can differentiate any polynomial term by term, in seconds.