Solve the Following Question.(3 Marks)

Question 13 Marks

Solve the following: : The demand $(D)$ of biscuits at price $P$ is given by $D=\frac{64}{P^3}$, find the marginal demand when the price is ₹ 4 /-

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Question 23 Marks

Solve the following: : The relation between price $(P)$ and demand $(D)$ of a cup of Tea is given by $D=\frac{12}{P}$. Find the rate at which the demand changes when the price is ₹ $2 /-$. Interpret the result.

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Question 33 Marks

Solve the following examples: : The cost of producing $x$ articles is given by $C=x^2+15 x+81$. Find the average cost and marginal cost functions. Find marginal cost when $x=10$. Find $x$ for which the marginal cost equals the average cost.

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Question 43 Marks

Solve the following examples: : If for a commodity; the demand function is given by, $D=\sqrt{75-3 P}$. Find the marginal demand function when $P=5$.

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Question 53 Marks

Solve the following examples: : The total cost function of producing $n$ notebooks is given by $C=1500-75 n+2 n^2+\frac{n^3}{5}$
Find the marginal cost at $n=10$.

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Question 63 Marks

Differentiate the following functions w.r.t. x. : $\frac{(x+1)(x-1)}{\left(e^x+1\right)}$

Answer

$
\begin{aligned}
& \text { Let } y=\frac{(x+1)(x-1)}{\left(\mathrm{e}^x+1\right)} \\
& y=\frac{x^2-1}{\left(\mathrm{e}^x+1\right)}
\end{aligned}
$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
\frac{\mathrm{d} y}{\mathrm{~d} x} & =\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{x^2-1}{\mathrm{e}^x+1}\right) \text { } \\
& =\frac{\left(\mathrm{e}^x+1\right) \frac{\mathrm{d}}{\mathrm{d} x}\left(x^2-1\right)-\left(x^2-1\right) \frac{\mathrm{d}}{\mathrm{d} x}\left(\mathrm{e}^x+1\right)}{\left(\mathrm{e}^x+1\right)^2} \\
& =\frac{\left(\mathrm{e}^x+1\right)(2 x)-\left(x^2-1\right)\left(\mathrm{e}^x+0\right)}{\left(\mathrm{e}^x+1\right)^2} \\
& =\frac{2 x \mathrm{e}^x+2 x-x^2 \mathrm{e}^x+\mathrm{e}^x}{\left(\mathrm{e}^x+1\right)^2} \\
& =\frac{2 x \mathrm{e}^x+\mathrm{e}^x-x^2 \mathrm{e}^x+2 x}{\left(\mathrm{e}^x+1\right)^2} \\
& =\frac{\mathrm{e}^x\left(2 x+1-x^2\right)+2 x}{\left(\mathrm{e}^x+1\right)^2}
\end{aligned}
$

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Question 73 Marks

Find the derivatives of the following functions by the first principle : \begin{equation}
\frac{1}{2 x+3}
\end{equation}

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Question 83 Marks

Find the derivatives of the following functions by the first principle : \begin{equation}
3 x^2+4
\end{equation}

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Question 93 Marks

Find the derivatives of the following w.r.t. x. : \begin{equation}
\frac{x \mathrm{e}^x}{x+\mathrm{e}^x}
\end{equation}

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Question 103 Marks

Find the derivatives of the following w.r.t. x. : \begin{equation}
\frac{3 e^x-2}{3 e^x+2}
\end{equation}

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Question 113 Marks

Find the derivatives of the following w.r.t. x. : \begin{equation}
\frac{\log x}{x^3-5}
\end{equation}

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Question 123 Marks

Find the derivatives of the following w.r.t. x. : \begin{equation}
\frac{3 x^2+5}{2 x^2-4}
\end{equation}

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Question 133 Marks

Find the derivatives of the following w.r.t. x. : \begin{equation}
\frac{x^2+a^2}{x^2-a^2}
\end{equation}

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Maths (commerce) Solve the Following Question.(3 Marks)

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