Balbharati Maharashtra State Board 11th Commerce Maths Solution Book Pdf Chapter 9 Differentiation Ex 9.1 Questions and Answers.

## Maharashtra State Board 11th Commerce Maths Solutions Chapter 9 Differentiation Ex 9.1

I. Find the derivatives of the following functions w.r.t. x.

Question 1.

x^{12}

Solution:

Let y = x^{12}

Differentiating w.r.t. x, we get

Question 2.

x^{-9}

Solution:

Let y = x^{-9}

Differentiating w.r.t. x, we get

Question 3.

\(x^{\frac{3}{2}}\)

Solution:

Let y = \(x^{\frac{3}{2}}\)

Differentiating w.r.t. x, we get

Question 4.

7x√x

Solution:

Question 5.

3^{5}

Solution:

Let y = 3^{5}

Differentiating w.r.t. x, we get

\(\frac{d y}{d x}=\frac{d}{d x} 3^{5}=0\) …..[3^{5} is a constant]

II. Differentiate the following w.r.t. x.

Question 1.

x^{5} + 3x^{4}

Solution:

Let y = x^{5} + 3x^{4}

Differentiating w.r.t. x, we get

Question 2.

x√x + log x – e^{x}

Solution:

Let y = x√x + log x – e^{x}

= \(x^{\frac{3}{2}}+\log x-e^{x}\)

Differentiating w.r.t. x, we get

Question 3.

\(x^{\frac{5}{2}}+5 x^{\frac{7}{5}}\)

Solution:

Let y = \(x^{\frac{5}{2}}+5 x^{\frac{7}{5}}\)

Differentiating w.r.t. x, we get

Question 4.

\(\frac{2}{7} x^{\frac{7}{2}}+\frac{5}{2} x^{\frac{2}{5}}\)

Solution:

Let y = \(\frac{2}{7} x^{\frac{7}{2}}+\frac{5}{2} x^{\frac{2}{5}}\)

Differentiating w.r.t. x, we get

Question 5.

\(\sqrt{x}\left(x^{2}+1\right)^{2}\)

Solution:

Let y = \(\sqrt{x}\left(x^{2}+1\right)^{2}\)

III. Differentiate the following w.r.t. x.

Question 1.

x^{3} log x

Solution:

Let y = x^{3} log x

Differentiating w.r.t. x, we get

Question 2.

\(x^{\frac{5}{2}} e^{x}\)

Solution:

Let y = \(x^{\frac{5}{2}} e^{x}\)

Differentiating w.r.t. x, we get

Question 3.

e^{x} log x

Solution:

Let y = e^{x} log x

Differentiating w.r.t. x, we get

Question 4.

x^{3} . 3^{x}

Solution:

Let y = x^{3} . 3^{x}

Differentiating w.r.t. x, we get

IV. Find the derivatives of the following w.r.t. x.

Question 1.

\(\frac{x^{2}+a^{2}}{x^{2}-a^{2}}\)

Solution:

Question 2.

\(\frac{3 x^{2}+5}{2 x^{2}-4}\)

Solution:

Question 3.

\(\frac{\log x}{x^{3}-5}\)

Solution:

Question 4.

\(\frac{3 e^{x}-2}{3 e^{x}+2}\)

Solution:

Question 5.

\(\frac{x \mathrm{e}^{x}}{x+\mathrm{e}^{x}}\)

Solution:

V. Find the derivatives of the following functions by the first principle:

Question 1.

3x^{2} + 4

Solution:

Let f(x) = 3x^{2} + 4

∴ f(x + h) = 3(x + h)^{2} + 4

= 3(x^{2} + 2xh + h^{2}) + 4

= 3x^{2} + 6xh + 3h^{2} + 4

By first principle, we get

Question 2.

x√x

Solution:

Let f(x) = x√x

∴ f(x + h) = \((x+h)^{\frac{3}{2}}\)

By first principle, we get

Question 3.

\(\frac{1}{2 x+3}\)

Solution:

Let f(x) = \(\frac{1}{2 x+3}\)

∴ f(x + h) = \(\frac{1}{2(x+\mathrm{h})+3}=\frac{1}{2 x+2 \mathrm{~h}+3}\)

By first principle, we get

Question 4.

\(\frac{x-1}{2 x+7}\)

Solution:

Let f(x) = \(\frac{x-1}{2 x+7}\)

∴ f(x + h) = \(\frac{x+\mathrm{h}-1}{2(x+\mathrm{h})+7}=\frac{x+\mathrm{h}-1}{2 x+2 \mathrm{~h}+7}\)

By first principle, we get