Balbharati Maharashtra State Board 12th Commerce Maths Solution Book Pdf

Chapter 3 Differentiation Ex 3.5 Questions and Answers.

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

Question 1.
x = at², y = 2at
Solution:
x = at², y = 2at
Differentiating x and y w.r.t. t, we get


Question 2.
x = 2at², y = at⁴
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 + u²)
Solution:
x = \(\sqrt{1+u^{2}}\), y = log(1 + u²) ……(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