# Differentials

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**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) |
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) |
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 |

**12th grade math**

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