Application of Derivatives

Rate of Change of Quantities


  • If a quantity y varies with another quantity x, satisfying some rule y = f(x), the \frac{dy}{dx} (or f'(x)) represents the rate of change of y with respect to x and \frac{dy}{dx}]_{x=x_0} (or f'(x0)) represents the rate of change of y with respect to x at x = x0.
  • If two variable x and y are varying with respect to another variable t, i.e., if x = f(t) and y = g(t), then by Chain Rule
    \frac{dy}{dx}=\frac{dy}{dt}/\frac{dy}{dt},\ \tt if\ \frac{dy}{dt}\neq 0
  • \lim_{x \rightarrow a}\frac{f(x)}{g(x)}=a finite number (≠ 0), then we can say that f(x) and g(x) are infinitesimals of same order.
  • \lim_{x \rightarrow a}\frac{f(x)}{g(x)}=0, then we can say that f(x) and g(x) are equivalent.
  • \lim_{x \rightarrow 0}\frac{f(x)}{g(x)}=0, then we can say that f(x) is higher order than g(x).
  • \lim_{x \rightarrow 0}\frac{f(x)}{g(x)}=\infty, then we can say that g(x) is higher order than f(x).

View the Topic in this video From 27:31 To 49:10

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

  • If s = f(t) is a relation between s (displacement) and t (time) of a particle and v = velocity =\frac{ds}{dt}
    a=\frac{dv}{dt}=\frac{d^{2}s}{dt^{2}}
    → v > 0 then s is increasing and particle moving from left to right
    → If v < 0 then s is decreasing and particle is moving from right to left
    → If v = 0 then particle comes to rest.
  • A particle is momentarily at rest \Rightarrow\frac{ds}{dt}=0,\frac{d^{2}s}{dt^{2}}\neq0
  • For maximum height of paritcle \frac{ds}{dt}=0
  • For maximum velocity \frac{d^{2}s}{dt^{2}}=0
  • A particle changes direction \Rightarrow\frac{ds}{dt}=0 \ and \ \frac{d^{2}s}{dt^{2}}\neq0