12th Commerce Maths 1 Chapter 3 Exercise 3.5 Answers Maharashtra Board

Differentiation Class 12 Commerce Maths 1 Chapter 3 Exercise 3.5 Answers Maharashtra Board

Balbharati Maharashtra State Board 12th Commerce Maths Solution Book Pdf Chapter 3 Differentiation Ex 3.5 Questions and Answers.

Std 12 Maths 1 Exercise 3.5 Solutions Commerce Maths

1. Find \(\frac{d y}{d x}\) if:

Question 1.
x = at ² , y = 2at
Solution:
x = at ² , y = 2at
Differentiating x and y w.r.t. t, we get

Question 2.
x = 2at ² , y = at ⁴
Solution:

Question 3.
x = e ^3t , y = e ^(4t+5)
Solution:
x = e ^3t , y = e ^(4t+5)
Differentiating x and y w.r.t. t, we get

2. Find \(\frac{d y}{d x}\) if:

Question 1.
x = \(\left(u+\frac{1}{u}\right)^{2}\), y = \((2)^{\left(u+\frac{1}{u}\right)}\)
Solution:
x = \(\left(u+\frac{1}{u}\right)^{2}\), y = \((2)^{\left(u+\frac{1}{u}\right)}\) ……(1)
Differentiating x and y w.r.t. u, we get,

Question 2.
x = \(\sqrt{1+u^{2}}\), y = log(1 + u ² )
Solution:
x = \(\sqrt{1+u^{2}}\), y = log(1 + u ² ) ……(1)
Differentiating x and y w.r.t. u, we get,

Question 3.
Differentiate 5 ^x with respect to log x.
Solution:
Let u = 5 ^x and v = log x
Then we want to find \(\frac{d u}{d v}\)
Differentiating u and v w.r.t. x, we get
\(\frac{d u}{d x}=\frac{d}{d x}\left(5^{x}\right)=5^{x} \cdot \log 5\)

3. Solve the following:

Question 1.
If x = \(a\left(1-\frac{1}{t}\right)\), y = \(a\left(1+\frac{1}{t}\right)\), then show that \(\frac{d y}{d x}\) = -1
Solution:

Question 2.
If x = \(\frac{4 t}{1+t^{2}}\), y = \(3\left(\frac{1-t^{2}}{1+t^{2}}\right)\), then show that \(\frac{d y}{d x}=-\frac{9 x}{4 y}\)
Solution:

Question 3.
If x = t . log t, y = t ^t , then show that \(\frac{d y}{d x}\) – y = 0.
Solution:
x = t log t
Differentiating w.r.t. t, we get