Differentials
Definition of Differentials
Let y = f(x) represent a function that is differentiable on an open interval
containing 'x'. The differential of x (denoted by dx) is any nonzero real number.
The differential of y (denoted by dy) is
dy = f '(x) dx
Sometimes differentials y can be used as approximation of change in y which is
$\triangle y \approx$ dy
OR
$\triangle y \approx$ f '(x) dx
In the earlier section, ask-math explains you about the tangent line approximation, which is
Consider any function that is differentiable at x = c so the equation of tangent line at (c,f(c)) is given by
y = f(c) + f '(c)(x - c )
the (x - c) quantity is called the change in x and it is denoted by $\triangle x$
When $\triangle x$ is small then change is y is denoted as $\triangle y$. Now the tangent line approximation will be
$\triangle y = f(c + \triangle x$) - f (c)
$\triangle y \approx f '(c)\triangle x$
Examples on Differentials
Function :
1) y = $x^{3}$
2) y = 2sin(x)
3) y = $\sqrt{x}$
4) y = 2x.cos(x)
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Derivative
1) $\frac{\text{d}y}{\text{d}x} =3x^{2}$
2) $\frac{\text{d}y}{\text{d}x}$ = 2cos(x)
3) $\frac{\text{d}y}{\text{d}x} = \frac{1}{2\sqrt{x}}$
4) $\frac{\text{d}y}{\text{d}x}$ = 2 cos(x) - 2x sin(x)
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Differentials
1) dy = 3$x^{2}$dx
2) dy = 2 cos(x)dx
3) dy = $\frac{dx}{2\sqrt{x}}$
4) dy = (2 cos(x) - 2x sin(x))dx
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12th grade math
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