Question

If y= 1+x/1!+x^2/2!+x^3/3!,
Prove that dy/dx=y

Answer 1

Answer:

Please see the attachment

Answer 2

Step-by-step explanation:

We know that,

eˣ = 1 + x/1! + x²/2! + x³/3! + x⁴/4! + ... ∞

Let, y = eˣ (as given)

Now differentiating both sides with respect to x, we get

dy/dx = d/dx (eˣ)

= eˣ

= y, as considered

Thus dy/dx = y. Hence proved.

Rules:

• d/dx (eᵐˣ) = m eᵐˣ, m is a rational number

• d/dx (xᵐ) = m xᵐ⁻¹, m is an integer

Note: However we can proceed by taking derivatives of the terms of x and we will get y again, as given. Thus we can prove dy/dx = y.

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