Question

Find the derivative of Sin√x using First principle.

Chapter = Limits and Derivatives
Class XI CBSE

Answer 1

Let the given Function be written as
f(x) = sin √x
f(x+h) = sin √x+h
Now,By first principle,
d(f(x))/dx = lim h->0 {f(x+h)-f(x)}/h
So,
d(sin√x)/dx = Lim h->0 sin√x+h - sin√x / h
=lim h->0 { 2 cos ( √x+h + √x / 2) sin ( √x+h - √x / 2) } / h
= Lim h->0 { 2 cos (√x+h + √x/2) sin ( √x+h-√x/2) / x + h - x
= lim h->0 { 2 cos (√x+h + √x/2) sin ( √x+h - √x/2) / {√x+h + √x}{√x+h-√x}
= Lim h->0 { 2 cos (√x+h + √x/2) sin(√x+h - √x/2) / 2(√x+h +√x) ( √x+h - √x)/2
= lim h->0 { cos ( √x+h + √x / 2) } / √x+h + √x
= cos (2√x/2) / √x + √x
= cos √x / 2√x

So,
d(sin√x)/dx = cos √x / 2√x
found by the first principle

Hope this helps you !

Answer 2

ANSWER

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