Rules of Differentiation

  • The ultimate reason of things must lie in a necessary substance, in which the differentiation of thechanges only exists eminently as in their source; and this is what we call God. (Leibniz)
  • The freedom of thought is a sacred right of every individual man, and diversity will continue to increase with the progress, refinement, and differentiation of the human intellect. (Felix Adler)
  • How did Biot arrive at the partial differential equation? [the heat conduction equation] . . .Perhaps Laplace gave Biot the equation and left him to sink or swim for a few years in trying to derive it. That would have been merely an instance of the way great mathematicians since the very beginnings of mathematical research have effortlessly maintained their superiority over ordinary mortals. (Clifford Truesdell)
  • Common integration is only the memory of differentiation. (Augustus De Morgan)

Derivative of a Linear Combination

We have directly from the definition

  • \frac{d}{dt}(u+v)=\frac{d(u+v)}{dt}=\frac{du}{dt}+\frac{dv}{dt}

and if \alpha is a constant

  • \frac{d}{dt}(\alpha u)=\frac{d(\alpha u)}{dt}=\alpha\frac{du}{dt}.

Combining these results we have

  • \frac{d}{dt}(\alpha u+\beta v)=\alpha\frac{du}{dt}+\beta\frac{dv}{dt}

where \alpha and \beta are constants.

Derivative of Product

If u(t) and v(t) are real-valued differentiable functions of t, then

  • d(uv)=udv+vdu

or

  • \frac{d}{dt}(uv)=\frac{du}{dt}v+u\frac{dv}{dt}=\dot uv+u\dot v,

because

  • \vert d(uv)-udv -vdu\vert =\vert u(t+dt)v(t+dt)-u(t)v(t)-u(t)\dot vdt-v\dot u dt\vert
  • = \vert u(t+dt)(v(t+dt)-v(t))-udv+v(t)(u(t+dt)-u(t))-vdu\vert
  • \le \vert u(t+dt)(v(t+dt)-v(t))-udv\vert +\vert v(t)(u(t+dt)-u(t))-vdu\vert \le C\vert dt\vert^2.

Derivative of a Quotient

We compute the derivative of the quotient \frac{1}{v(t)} assuming v(t)\neq 0 is differentiable:

  • \frac{1}{v(t+dt)}-\frac{1}{v(t)}=-\frac{v(t+dt)-v(t)}{v(t+dt)v(t)}\approx -\frac{\dot v(t)}{u(t)^2}dt

up to a quadratic deviation in \vert dt\vert. Thus, by combination with the previous result,

  • \frac{d}{dt}\frac{u}{v}=\frac{\dot uv-u\dot v}{v^2}.

The Chain Rule

If v:R\rightarrow R and w:R\rightarrow R are two differentiable functions, then the composite function u(x)=v(w(x)) is differentiable with derivative

  • u^\prime (x)=v^\prime (w(x))w^\prime (x)\quad\mbox{or}\quad\frac{du}{dx}=\frac{dv}{dw}\frac{dw}{dx}}.

To see this we estimate

  • u(x+dx)-u(x)=v(w(x+dx)-v(w(x))\approx v^\prime (w(x))(w(x+dx)-w(x))\approx v^\prime (v(x))w^\prime (x)dx

up to terms of order \vert dx\vert^2.

Read More

  • \hyperref[diffrules]{Rules of Differentiation}
  • \hyperref[intprops]{Rules of Integration}
  • Differentiation Rules

To Think About

  • How sensitive is differentiation to perturbations of function values?
  • How to compute thye derivate of a function if analytic differentiation is not an option, because it is too difficult or the function is not given by an analytic expression?

Watch

Leibniz’s first paper on calculus, Acta Eruditorum, 1684, with the above rules for differentiation.

 

Next: Proof of Fundamental Theorem    Previous: Derivative with respect to Space

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