Rules for Differentiation

Excerpts

• Rules for Differentiation

  This term, in theory, is the only one on which the *limit* of the ratio of the increments depends; in practice, it is sufficiently near to the real increment of $y$, if the increment of $x$ be small.

• Rules for Differentiation

  When the exponent is negative, or when $y = \dfrac{1}{x^{m}}$, $dy = -\dfrac{m\, dx}{x^{m+1}}$, or when $y = x^{-m}$, $dy = -mx^{-m-1}\, dx$, which is according to the rule.

• Rules for Differentiation

  The negative sign indicates that an increase in $x$ decreases the value of $y$; which, in this case, is evident.

• Rules for Differentiation

  (2) $y = a^{x}$. Here $dy = a^{x}\log a\, dx$ where the logarithm (as is always the case in analysis, except where the contrary is specially mentioned) is the Naperian or hyperbolic logarithm.

• Rules for Differentiation

  $y = \log x$ (the Naperian logarithm). Here $dy = \dfrac{dx}{x}$.

Equations

• Rules for Differentiation

  dy = mx^{m-1}\, dx

  For y = x^m with m whole or fractional, positive or negative, the differential dy equals m x^(m-1) dx.

• Rules for Differentiation

  dy = -\dfrac{m\, dx}{x^{m+1}}

  When y = 1/x^m, the differential dy equals minus m dx divided by x^(m+1), the negative sign showing that increasing x decreases y.

• Rules for Differentiation

  dy = -mx^{-m-1}\, dx

  When y = x^(-m), the differential dy equals minus m x^(-m-1) dx, consistent with the power rule.

• Rules for Differentiation

  dy = a^{x}\log a\, dx

  For y = a^x, the differential dy equals a^x times the natural logarithm of a, times dx.

• Rules for Differentiation

  dy = e^{x}\, dx

  For y = e^x, where e is the base of the Naperian logarithms, the differential dy equals e^x dx.

• Rules for Differentiation

  a = 2.7182818 = e

  The number whose Naperian logarithm is 1 is taken to be the base e of the hyperbolic logarithms, approximately 2.7182818.

• Rules for Differentiation

  dy = \dfrac{dx}{x}

  For y equal to the Naperian logarithm of x, the differential dy equals dx divided by x.

• Rules for Differentiation

  dy = -.4342944\, \dfrac{dx}{x}

  For y equal to the common logarithm of x, the differential dy equals minus 0.4342944 times dx divided by x; the constant is the decimal modulus 1/ln 10 given to seven places.

• Rules for Differentiation

  dy = \cos x\, dx

  For y equal to sin x, the differential dy equals cos x times dx.

• Rules for Differentiation

  dy = -\sin x\, dx

  For y equal to cos x, the differential dy equals minus sin x times dx.

• Rules for Differentiation

  dy = \dfrac{dx}{\cos^{2} x}

  For y equal to tan x, the differential dy equals dx divided by the square of cos x.

Problems

No exercises in this chapter.