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

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  • If s = f(t) is a relation between s (displacement) and t (time) of a particle and v = velocity =\frac{ds}{dt}
    → 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