\(y=\begin{vmatrix} f(x)&g(x) &h(x) \\ l&m &n \\ a&b &c \end{vmatrix}\),prove that \(\frac{dy}{dx}=\begin{vmatrix} f'(x)&g'(x) &h'(x) \\ l&m &n \\ a&b &c \end{vmatrix}\)
\(y=\begin{vmatrix} f(x)&g(x) &h(x) \\ l&m &n \\ a&b &c \end{vmatrix}\),prove that \(\frac{dy}{dx}=\begin{vmatrix} f'(x)&g'(x) &h'(x) \\ l&m &n \\ a&b &c \end{vmatrix}\)
Solution and Explanation
\(y=\begin{vmatrix} f(x)&g(x) &h(x) \\ l&m &n \\ a&b &c \end{vmatrix}\)
⇒ y=(mc-nb)f(x)-(lc-na)g(x)+(lb-ma)h(x)
Then,\(\frac{dy}{dx}\)=\(\frac{d}{dx}\)[(mc-nb)f(x)]-\(\frac{d}{dx}\)(lc-na)g(x)]+\(\frac{d}{dx}\)[(lb-ma)h(x)]
=(mc-nb)f'(x)-(lc-na)g'(x)+(lb-ma)h'(x)
=\(\begin{vmatrix} f'(x)&g'(x) &h'(x) \\ l&m &n \\ a&b &c \end{vmatrix}\)
Thus, \(\frac{dy}{dx}\)=\(\begin{vmatrix} f(x)&g(x) &h(x) \\ l&m &n \\ a&b &c \end{vmatrix}\)
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Notes on Continuity and differentiability
Concepts Used:
Continuity & Differentiability
Definition of Differentiability
f(x) is said to be differentiable at the point x = a, if the derivative f ‘(a) be at every point in its domain. It is given by
Definition of Continuity
Mathematically, a function is said to be continuous at a point x = a, if
It is implicit that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=a exist and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is unspecified or does not exist, then we say that the function is discontinuous.