Chapter 1 Differentiation Ex 1.3
Chapter 1 Differentiation Ex 1.3
Question 1.
Differentiate the following w.r.t. x:
Then log y = log [latex]\sqrt[3]{\frac{4 x-1}{(2 x+3)(5-2 x)^{2}}}[/latex]
Differentiating both sides w.r.t. x, we get
Then log y = log [latex]\left(x^{2}+3\right)^{\frac{3}{2}} \cdot \sin ^{3} 2 x \cdot 2^{x^{2}}[/latex]
Differentiating both sides w.r.t. x, we get
Then log y = log [latex]\frac{\left(x^{2}+2 x+2\right)^{\frac{3}{2}}}{(\sqrt{x}+3)^{3}(\cos x)^{x}}[/latex]
Differentiating both sides w.r.t. x, we get
Then log y = log [latex]\frac{x^{5} \cdot \tan ^{3} 4 x}{\sin ^{2} 3 x}[/latex]
= log + log ta4x – log si3x
= 5 log x+ 3 log (tan 4x) – 2 log (sin 3x)
Differentiating both sides w.r.t. x, we get
(vi)
Solution:
Let y =
Then log y = log () = ()(log x)
Differentiating both sides w.r.t. x, we get
(vii) (sin x
Solution:
Let y = (sin x
Then log y = log (sin x = x . log (sin x)
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Question 2.
Differentiate the following w.r.t. x:
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Question 3.
(i) √x + √y = √a
Solution:
√x + √y = √a
Differentiating both sides w.r.t. x, we get
(ii) x√x + y√y = a√a
Solution:
x√x + y√y = a√a
Differentiating both sides w.r.t. x, we get
(iii) x + √xy + y = 1
Solution:
x + √xy + y = 1
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
(vi)
Solution:
Differentiating both sides w.r.t. x, we get
(vii) = cos (x – y)
Solution:
= cos (x – y)
Differentiating both sides w.r.t. x, we get
(viii) cos (xy) = x + y
Solution:
cos (xy) = x + y
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Question 4.
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Solution:
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Solution:
Differentiating both sides w.r.t. x, we get
Question 5.
(i) If log (x + y) = log (xy) + p, where p is a constant, then prove that
Solution:
log (x + y) = log (xy) + p
∴ log (x + y) = log x + log y + p
Differentiating both sides w.r.t. x, we get
Solution:
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
Differentiating both sides w.r.t. x, we get
