"How do we find the differential of y=x2+1..." was answered by our experts

Freddy Novak

Answered question

2022-04-17

How do we find the differential of

y = x^{2} + 1

from first principle?

chambasos6

Beginner2022-04-18Added 12 answers

According to fist principle, if y=f(x)
then

\frac{d y}{d x} = L t_{h \to 0} \frac{f (x + h) - f (x)}{h}

Here

f (x) = {(x - 1)}^{2} + 1

therefore

f (x + h) = {(x + h - 1)}^{2} + 1

and f(x+h)-f(x)

= {(x + h - 1)}^{2} + 1 - {(x - 1)}^{2} + 1

= {(x + h - 1)}^{2} - {(x - 1)}^{2}

=(x+h-1+x-1)(x+h-1-x+1)
-using

a^{2} - b^{2} = (a + b) (a - b)

=h(2x+h-2)
and

\frac{d y}{d x} = L t_{h \to 0} \frac{h (2 x + h - 2)}{h}

= L t_{h \to 0} (2 x + h - 2)

=2x-2

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