Question 3. Differentiate each of the following using first principles:
(i) xsinx
Solution:
Given that f(x) = xsinx
By using the formula
We get
=
=
Using the formula
sinc – sind = 2cos((c + d)/2)sin((c – d)/2)
We get
=
As we know that
So,
= 2x × cosx × 1/2 + sinx
= x × cosx + sinx
= sinx + xcosx
(ii) xcosx
Solution:
Given that f(x) = xcosx
By using the formula
We get
=
=
=
=
=
= -xsinx + cosx
(iii) sin(2x – 3)
Solution:
Given that f(x) = sin(2x – 3)
By using the formula
We get
=
=
Using the formula
sinC – sinD = 2cos{C+D}/2sin{C-D}/2
=
As we know that, \lim_{θ\to 0}\frac{sinθ}{θ}=1 so,
= 2cos(2x – 3)
(iv) √sin2x
Solution:
Given that f(x) = √sin2x
By using the formula
We get
=
On multiplying numerator and denominator by
we get
=
=
=
=
=
(v) sinx/x
Solution:
Given that f{x} = sinx/x
By using the formula
We get
=
=
=
=
=
h ⇢ 0 ⇒ h/2 ⇢ 0 and
=
=
(vi) cosx/x
Solution:
Given that f(x) = cosx/x
By using the formula
We get
=
=
=
=
=
=
=
(vii) x2sinx
Solution:
Given that f(x) = x2sinx
By using the formula
We get
=
=
=
=
= 0 + [2xsinx + x2cosx]
= 2xsinx + x2cosx
(viii)
Solution:
Given that f(x) =
By using the formula
We get
=
=
=
=
=
(ix) sinx + cosx
Solution:
Given that f(x) = sinx + cosx
By using the formula
We get
=
=
=
=
=
=
= cosx – sinx
Question 4. Differentiate each of the following using first principles:
(i) tan2x
Solution:
Given that f(x) = tan2x
By using the formula
We get
=
=
=
=
=
=
=
= 2tanx sec2x
(ii) tan(2x + 1)
Solution:
Given that f(x) = tan(2x+1)
By using the formula
We get
=
=
=
Multiplying both, numerator and denominator by 2.
=
=
= 2sec2(2x+1)
(iii) tan2x
Solution:
Given that f(x) = tan2x
By using the formula
We get
=
=
=
=
=
= 2sec22x
(iv) √tanx
Solution:
Given that f(x) = √tanx
By using the formula
We get
=
On multiplying numerator and denominator by
We get
=
=
=
=
=
Question 5. Differentiate each of the following using first principles:
(i)
Solution:
Given that f(x) =
By using the formula
We get
=
=
=
=
=
=
(ii) cos√x
Solution:
Given that f(x) = cos√x
By using the formula
We get
=
=
=
Multiplying numerator and denominator by
=
=
=
=
(iii) tan√x
Solution:
Given that f(x) = tan√x
By using the formula
We get
=
=
=
=
=
=
=
=
(iv) tanx2
Solution:
Given that f(x) = tanx2
By using the formula
We get
=
=
=
=
=
=
=
=
= 2xsec2x2
Question 6. Differentiate each of the following using first principles:
(i) -x
Solution:
Given that f(x) = -x
By using the formula
We get
=
=
= -1
(ii) (-x)-1
Solution:
Given that f(x) = (-x)-1
By using the formula
We get
=
=
=
= 1/x2
(iii) sin(x + 1)
Solution:
Given that f(x) = sin(x+1)
By using the formula
We get
=
=
=
=
=
= cos(x+1)
(iv) cos(x – π/8)
Solution:
We have, f(x) = cos(x – π/8)
By using the formula
We get
=
=
=
=
=
=
= -sin(x + π/8)