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

## Maharashtra State Board 12th Commerce Maths Solutions Chapter 3 Differentiation Ex 3.5

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