Solve the following Question.(1 Marks)

Question 11 Mark

Solve the following: : The price function $P$ of a commodity is given as $P=20+D-D^2$ where $D$ is demand. Find the rate at which price $(P)$ is changing when demand $D=3$.

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Question 21 Mark

Find $\frac{d y}{d x}$ if : $y=(1-x)(2-x)$

Answer

$
\begin{aligned}
y & =(1-x)(2-x) \\
& =2-3 x+x^2
\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(2-3 x+x^2\right) \\
& =\frac{\mathrm{d}}{\mathrm{d} x}(2)-3 \frac{\mathrm{d}}{\mathrm{d} x}(x)+\frac{\mathrm{d}}{\mathrm{d} x}\left(x^2\right) \\
& =0-3(1)+2 x \\
& =-3+2 x
\end{aligned}
$

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Question 31 Mark

Find $\frac{d y}{d x}$ if : $y=x^2+2 x-1$

Answer

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

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Question 41 Mark

Differentiate the following functions w.r.t.x. : $7^x$

Answer

Let $y=7^x$
Differentiating w.r.t. $x$, we get
$
\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d}}{\mathrm{d} x} 7^x=7^x \log 7
$

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Question 51 Mark

Differentiate the following functions w.r.t.x. : $\frac{1}{\sqrt{x}}$

Answer

$
\begin{aligned}
& \text { Let } \mathrm{y}=\frac{1}{\sqrt{x}} \\
& \therefore \mathrm{y}=x^{\frac{-1}{2}}
\end{aligned}
$
Differentiating w.r.t. $\mathrm{x}$, we get
$
\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{-1}{2} x^{\frac{-3}{2}}=\frac{-1}{2 x^{\frac{3}{2}}}
$

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Question 61 Mark

Differentiate the following functions w.r.t.x. : $\mathrm{x} \sqrt{\mathrm{x}}$

Answer

$
\begin{aligned}
& \text { Let } \mathrm{y}=\mathrm{x} \sqrt{\mathrm{x}} \\
& \therefore \mathrm{y}=x^{\frac{3}{2}}
\end{aligned}
$
Differentiating w.r.t. $x$, we get
$
\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d}}{\mathrm{d} x} x^{\frac{3}{2}}=\frac{3}{2} x^{\frac{1}{2}}
$

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Question 71 Mark

Differentiate the following functions w.r.t.x. : $\sqrt{x}$

Answer

Let $y=\sqrt{ } x$
Differentiating w.r.t. $x$, we get
$
\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d}}{\mathrm{d} x} \sqrt{x}=\frac{1}{2 \sqrt{x}}
$

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Question 81 Mark

Differentiate the following functions w.r.t.x. : $x^{-2}$

Answer

Let $y=x^{-2}$
Differentiating w.r.t. $x$, we get
$
\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d}}{\mathrm{d} x}\left(x^{-2}\right)=-2 x^{-3}=\frac{-2}{x^3}
$

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Question 91 Mark

Differentiate the following functions w.r.t.x.
$x^5$

Answer

Let $y=x^5$
Differentiating w.r.t. $x$, we get
$
\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d}}{\mathrm{d} x} x^5=5 x^4
$

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Question 101 Mark

Solve the following examples: : The demand function of a commodity is given as $P=20+D-D^2$. Find the rate at which price is changing when demand is 3 .

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Question 111 Mark

Solve the following examples: : If for a commodity; the price-demand relation is given as $D=\frac{P+5}{P-1}$. Find the marginal demand when the price is 2 .

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Question 121 Mark

Differentiate the following functions w.r.t. x. : $\frac{2^x}{\log x}$

Answer

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

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Question 131 Mark

Differentiate the following functions w.r.t. x. : $\frac{x}{\log x}$

Answer

Let $y=\frac{x}{\log x}$
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}{\log x}\right)_{\text { }} \\
& =\frac{\log x \frac{\mathrm{d}}{\mathrm{d} x}(x)-x \frac{\mathrm{d}}{\mathrm{d} x}(\log x)}{(\log x)^2} \\
& =\frac{\log x(1)-x\left(\frac{1}{x}\right)}{(\log x)^2} \\
& =\frac{\log x-1}{(\log x)^2}
\end{aligned}
$

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Question 141 Mark

Differentiate the following w.r.t. x. : \begin{equation}
x^3 \cdot 3^x
\end{equation}

Answer

Let $y=x^3 \cdot 3^x$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
\frac{\mathrm{d} y}{\mathrm{~d} x} & =\frac{\mathrm{d}}{\mathrm{d} x}\left(x^3 3^x\right) \\
& =x^3 \frac{\mathrm{d}}{\mathrm{d} x}\left(3^x\right)+3^x \frac{\mathrm{d}}{\mathrm{d} x}\left(x^3\right) \\
& =\left(x^3\right)\left(3^x \log 3\right)+3^x\left(3 x^2\right) \\
& =x^2 3^x(x \log 3+3)
\end{aligned}
$

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Question 151 Mark

Differentiate the following w.r.t. x. : \begin{equation}
x^3 \log x
\end{equation}

Answer

Let $y=x^3 \log x$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
\frac{\mathrm{d} y}{\mathrm{~d} x} & =\frac{\mathrm{d}}{\mathrm{d} x} x^3 \log x \\
& =x^3 \frac{\mathrm{d}}{\mathrm{d} x}(\log x)+(\log x) \frac{\mathrm{d}}{\mathrm{d} x}\left(x^3\right) \\
& =x^3 \times \frac{1}{x}+(\log x)\left(3 x^2\right) \\
& =x^2+3 x^2 \log x
\end{aligned}
$

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Question 161 Mark

Differentiate the following w.r.t. x. : $x^5+3 x^4$

Answer

Let $y=x^5+3 x^4$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
\frac{\mathrm{d} y}{\mathrm{~d} x} & =\frac{\mathrm{d}}{\mathrm{d} x}\left(x^5+3 x^4\right) \\
& =\frac{\mathrm{d}}{\mathrm{d} x} x^5+3 \frac{\mathrm{d}}{\mathrm{d} x} x^4 \\
& =5 x^4+3\left(4 x^3\right) \\
\frac{\mathrm{d} y}{\mathrm{~d} x} & =5 x^4+12 x^3
\end{aligned}
$

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Question 171 Mark

Find the derivatives of the following functions w.r.t. x. : $
3^5
$

Answer

Let $y=3^5$
Differentiating w.r.t. $x$, we get $\frac{d y}{d x}=\frac{d}{d x} 3^5=0 \ldots . .\left[3^5\right.$ is a constant $]$

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Question 181 Mark

Find the derivatives of the following functions w.r.t. x. : $
7 \mathrm{x} \sqrt{\mathrm{x}}
$

Answer

Let $y=7 x \sqrt{x}$
$
\begin{aligned}
& =7 x^1 x^{\frac{1}{2}} \\
y & =7 x^{\frac{3}{2}}
\end{aligned}
$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
\frac{\mathrm{d} y}{\mathrm{~d} x} & =\frac{\mathrm{d}}{\mathrm{d} x} 7 x^{\frac{3}{2}} \\
& =7 \times \frac{3}{2} x^{\frac{3}{2}-1} \\
& =\frac{21}{2} x^{\frac{1}{2}} \\
& =\frac{21}{2} \sqrt{x}
\end{aligned}
$

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Question 191 Mark

Find the derivatives of the following functions w.r.t. x. : $
x^{\frac{3}{2}}
$

Answer

Let $\mathrm{y}=x^{\frac{3}{2}}$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
\frac{\mathrm{d} y}{\mathrm{~d} x} & =\frac{\mathrm{d}}{\mathrm{d} x} x^{\frac{3}{2}} \\
& =\frac{3}{2} x^{\frac{3}{2}-1} \\
& =\frac{3}{2} x^{\frac{1}{2}} \\
& =\frac{3}{2} \sqrt{x}
\end{aligned}
$

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Question 201 Mark

Find the derivatives of the following functions w.r.t. x.
$x^{-9}$

Answer

Let $y=x^{-9}$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
\frac{\mathrm{d} y}{\mathrm{~d} x} & =\frac{\mathrm{d}}{\mathrm{d} x} x^{-3} \\
& =-9 x^{-9-1} \\
& =-9 x^{-10}
\end{aligned}
$

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Question 211 Mark

Find the derivatives of the following functions w.r.t. x. : x^{12}

Answer

Let $y=x^{12}$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
\frac{\mathrm{d} y}{\mathrm{~d} x} & =\frac{\mathrm{d}}{\mathrm{d} x} x^{12} \\
& =12 x^{12-1} \\
& =12 x^{11}
\end{aligned}
$

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Maths (commerce) Solve the following Question.(1 Marks)

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