DIFFERENTIABILITY

DIFFERENTIABILITY

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DIFFERENTIABILITY

3) Why?
Proof:
Let y = f₁(x) + f₂(x), Where f₁(x) and f₂(x) are differentiable functions of x.
Then

Thus .
4) Why?
Proof:
Let y = f₁(x).f₂(x) where both f₁(x) and f₂(x) are differentiable function of x.

5) Why?
Proof:
Let , where f₁(x) and f₂(x) are differentiable functions at all points in its domain and f₂(x) ¹ 0.

6) Why?

Proof:
Let y = f (t), t = g(x)
Then y = f (g(x)) is a function of x.
Dy = f (t+Dt) - f (t)
And Dt = g (x+Dx) - g(x)

Assuming that for sufficient small values of Dx, Dt ¹ 0 this will necessarily be the case if then since g is differentiable it is continuous and hence when Dx®0, x+Dx®x any g(x+Dx) ® g(x).
Therefore t+Dt®t and Dt®0, since Dt ¹ 0 and Dx®0 we may write,
.
Clearly f and g are both continuous functions because they are differentiable thus Dt®0 when Dx®0 and Dy®0 when Dt®0.

Hence

Standard Formulae of Differentiation:
1)
Proof:
Let

Thus
2)
Proof:

Thus
3)
Proof:

So,
4)
Proof: