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Second Year Math Differentiation 4. Find the derivative w.r.t. x (ii) \sin \sqrt{\frac{1+2 x}{1+x}}


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4. Find the derivative w.r.t. x (ii) \sin \sqrt{\frac{1+2 x}{1+x}}

1. Apply the Maclaurin series expansion to prove that:(ii) \cos x=1-\frac{x^{2}}{2}+\frac{x^{4}}{4}-\frac{x^{6}}{16}+\ldots \ldots
1. Apply the Maclaurin series expansion to prove that:(ii)  \cos x=1-\frac{x^{2}}{2}+\frac{x^{4}}{4}-\frac{x^{6}}{16}+\ldots \ldots

1. Apply the Maclaurin series expansion to prove that:(ii) \cos x=1-\frac{x^{2}}{2}+\frac{x^{4}}{4}-\frac{x^{6}}{16}+\ldots \ldots

1. Find by definition the derivatives w.r.t x of the following functions defined as:\[\text { (xi) } x^{m} m \in N\]
1. Find by definition the derivatives w.r.t   x   of the following functions defined as:\[\text { (xi) }  x^{m} m \in N\]

1. Find by definition the derivatives w.r.t x of the following functions defined as:\[\text { (xi) } x^{m} m \in N\]

Find the derivative of \sqrt{x} at x=a from first principle
Find the derivative of  \sqrt{x}  at  x=a  from first principle

Find the derivative of \sqrt{x} at x=a from first principle

Example 5: Find the derivative of x^{3}+2 x+3
Example 5: Find the derivative of  x^{3}+2 x+3

Example 5: Find the derivative of x^{3}+2 x+3

3. Find \frac{d y}{d x} if(iii) y=\tanh ^{-1}(\sin x) \frac{\pi}{2} x \frac{\pi}{2}
3. Find  \frac{d y}{d x}  if(iii)  y=\tanh ^{-1}(\sin x)  \frac{\pi}{2}  x  \frac{\pi}{2}

3. Find \frac{d y}{d x} if(iii) y=\tanh ^{-1}(\sin x) \frac{\pi}{2} x \frac{\pi}{2}

Example 3: Find \frac{d y}{d x} if y=\frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}} (x \neq 0)
Example  3: Find  \frac{d y}{d x}  if  y=\frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}}  (x \neq 0)

Example 3: Find \frac{d y}{d x} if y=\frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}} (x \neq 0)

2. Find \frac{d y}{d x} if:(v) x \sqrt{1+y}+y \sqrt{1+x}=0
2. Find  \frac{d y}{d x}  if:(v)   x \sqrt{1+y}+y \sqrt{1+x}=0

2. Find \frac{d y}{d x} if:(v) x \sqrt{1+y}+y \sqrt{1+x}=0

5. Differentiate(i) x^{2}-\frac{1}{x^{2}} w.r.t x^{4} (ii) \left(1+x^{2}\right)^{n} w.r.t x^{2} (iii) \frac{x^{2}+1}{x^{2}-1} w.r.t \frac{x-1}{x+1} (iv) \frac{a x+b}{c x+d} w \cdot r \cdot t \frac{a x^{2}+b}{a x^{2}+d} (v) \frac{x^{2}+1}{x^{2}-1} w.r.t x^{3}
5. Differentiate(i)  x^{2}-\frac{1}{x^{2}}  w.r.t  x^{4} (ii)  \left(1+x^{2}\right)^{n}  w.r.t  x^{2} (iii)  \frac{x^{2}+1}{x^{2}-1}  w.r.t  \frac{x-1}{x+1} (iv)  \frac{a x+b}{c x+d} w \cdot r \cdot t \frac{a x^{2}+b}{a x^{2}+d} (v)  \frac{x^{2}+1}{x^{2}-1}  w.r.t  x^{3}

5. Differentiate(i) x^{2}-\frac{1}{x^{2}} w.r.t x^{4} (ii) \left(1+x^{2}\right)^{n} w.r.t x^{2} (iii) \frac{x^{2}+1}{x^{2}-1} w.r.t \frac{x-1}{x+1} (iv) \frac{a x+b}{c x+d} w \cdot r \cdot t \frac{a x^{2}+b}{a x^{2}+d} (v) \frac{x^{2}+1}{x^{2}-1} w.r.t x^{3}

12. Find the point on the curve y=x^{2}+1 that is closest to the point (181) .
12. Find the point on the curve  y=x^{2}+1  that is closest to the point  (181) .

12. Find the point on the curve y=x^{2}+1 that is closest to the point (181) .

1. Find from first principles the derivatives of the following expressions w.r.t. their respective independent variables:(ii) (2 x+3)^{5}
1. Find from first principles the derivatives of the following expressions w.r.t. their respective independent variables:(ii)  (2 x+3)^{5}

1. Find from first principles the derivatives of the following expressions w.r.t. their respective independent variables:(ii) (2 x+3)^{5}

1. Apply the Maclaurin series expansion to prove that:(v) e^{2 x} =1+2 x+\frac{4 x^{2}}{\lfloor 2}+\frac{8 x^{3}}{\lfloor} \underline{3}+\ldots \ldots
1. Apply the Maclaurin series expansion to prove that:(v)   e^{2 x} =1+2 x+\frac{4 x^{2}}{\lfloor 2}+\frac{8 x^{3}}{\lfloor} \underline{3}+\ldots \ldots

1. Apply the Maclaurin series expansion to prove that:(v) e^{2 x} =1+2 x+\frac{4 x^{2}}{\lfloor 2}+\frac{8 x^{3}}{\lfloor} \underline{3}+\ldots \ldots

2. Find the extreme values for the following functions defined as:\[\text { (viii) } f(x)=(x-2)^{2}(x-1)\]
2. Find the extreme values for the following functions defined as:\[\text { (viii) } f(x)=(x-2)^{2}(x-1)\]

2. Find the extreme values for the following functions defined as:\[\text { (viii) } f(x)=(x-2)^{2}(x-1)\]

Example 2: What are the dimensions of a box of a square base having largest volume if the sum of one side of the base and its height is 12 \mathrm{~cm} .
Example 2: What are the dimensions of a box of a square base having largest volume if the sum of one side of the base and its height is  12 \mathrm{~cm} .

Example 2: What are the dimensions of a box of a square base having largest volume if the sum of one side of the base and its height is 12 \mathrm{~cm} .

Example 3:Expand a^{x} in the Maclaurin series.
Example 3:Expand  a^{x}  in the Maclaurin series.

Example 3:Expand a^{x} in the Maclaurin series.

Example 5:Find the first four derivatives of \cos (a x+b) .
Example 5:Find the first four derivatives of  \cos (a x+b) .

Example 5:Find the first four derivatives of \cos (a x+b) .

1. Find f^{\prime}(x) if(iv) f(x)=\frac{e^{x}}{e^{-x}+1}
1. Find  f^{\prime}(x)  if(iv)  f(x)=\frac{e^{x}}{e^{-x}+1}

1. Find f^{\prime}(x) if(iv) f(x)=\frac{e^{x}}{e^{-x}+1}

6. Find the lengths of the sides of a variable rectangle having area 36 \mathrm{~cm}^{2} when its perimeter is minimum.
6. Find the lengths of the sides of a variable rectangle having area  36 \mathrm{~cm}^{2}  when its perimeter is minimum.

6. Find the lengths of the sides of a variable rectangle having area 36 \mathrm{~cm}^{2} when its perimeter is minimum.

2. Find the extreme values for the following functions defined as:(iii) f(x)=5 x^{2}-6 x+2
2. Find the extreme values for the following functions defined as:(iii)  f(x)=5 x^{2}-6 x+2

2. Find the extreme values for the following functions defined as:(iii) f(x)=5 x^{2}-6 x+2

Example 7: If y=\operatorname{Sin}^{-1} \frac{x}{a} then show that y_{2} x\left(\begin{array}{ll}a^{2} & x^{2}\end{array}\right)
Example 7: If  y=\operatorname{Sin}^{-1} \frac{x}{a}  then show that  y_{2}  x\left(\begin{array}{ll}a^{2} & x^{2}\end{array}\right)
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Example 7: If y=\operatorname{Sin}^{-1} \frac{x}{a} then show that y_{2} x\left(\begin{array}{ll}a^{2} & x^{2}\end{array}\right)

Find higher derivatives of the polynomial\[f(x)=\frac{1}{12} x^{4}-\frac{1}{6} x^{3}+\frac{1}{4} x^{2}+2 x+7\]
Find higher derivatives of the polynomial\[f(x)=\frac{1}{12} x^{4}-\frac{1}{6} x^{3}+\frac{1}{4} x^{2}+2 x+7\]
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Find higher derivatives of the polynomial\[f(x)=\frac{1}{12} x^{4}-\frac{1}{6} x^{3}+\frac{1}{4} x^{2}+2 x+7\]

4. Find the derivative w.r.t. x (ii) \sin \sqrt{\frac{1+2 x}{1+x}}
4. Find the derivative w.r.t.  x (ii)  \sin \sqrt{\frac{1+2 x}{1+x}}
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4. Find the derivative w.r.t. x (ii) \sin \sqrt{\frac{1+2 x}{1+x}}

2. Find \frac{d y}{d x} if(ii) y=x \sqrt{\ln x}
2. Find  \frac{d y}{d x}  if(ii)  y=x \sqrt{\ln x}
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2. Find \frac{d y}{d x} if(ii) y=x \sqrt{\ln x}

2. Find \frac{d y}{d x} if(i) y=x^{2} \ln \sqrt{x}
2. Find  \frac{d y}{d x}  if(i)  y=x^{2} \ln \sqrt{x}
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2. Find \frac{d y}{d x} if(i) y=x^{2} \ln \sqrt{x}

Example 2: Use the Taylor series expansion to find the value of \sin 31^{\circ} .
Example 2: Use the Taylor series expansion to find the value of  \sin 31^{\circ} .
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Example 2: Use the Taylor series expansion to find the value of \sin 31^{\circ} .

2. Find \frac{d y}{d x} if:(i) 3 x+4 y+7=0
2. Find  \frac{d y}{d x}  if:(i)   3 x+4 y+7=0
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2. Find \frac{d y}{d x} if:(i) 3 x+4 y+7=0

3. Show that 2^{x+h}=2^{x}\left\{1+(\ln 2) h+\frac{(\ln 2)^{2} h^{2}}{\lfloor 2}+\frac{(\ln 2)^{3} h^{3}}{\lfloor 3}+\ldots\right\}
3. Show that  2^{x+h}=2^{x}\left\{1+(\ln 2) h+\frac{(\ln 2)^{2} h^{2}}{\lfloor 2}+\frac{(\ln 2)^{3} h^{3}}{\lfloor 3}+\ldots\right\}
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3. Show that 2^{x+h}=2^{x}\left\{1+(\ln 2) h+\frac{(\ln 2)^{2} h^{2}}{\lfloor 2}+\frac{(\ln 2)^{3} h^{3}}{\lfloor 3}+\ldots\right\}

2. Find \frac{d y}{d x} if(iv) y=x^{2} \ln \frac{1}{x}
2. Find  \frac{d y}{d x}  if(iv)  y=x^{2} \ln \frac{1}{x}
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2. Find \frac{d y}{d x} if(iv) y=x^{2} \ln \frac{1}{x}

2. Find the extreme values for the following functions defined as:(iv) f(x)=3 x^{2}
2. Find the extreme values for the following functions defined as:(iv)  f(x)=3 x^{2}
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2. Find the extreme values for the following functions defined as:(iv) f(x)=3 x^{2}

1. Find \frac{d y}{d x} by making suitable substitutions in the following functions defined as:(v) \sqrt{\frac{a^{2}+x^{2}}{a^{2}-x}}
1. Find  \frac{d y}{d x}  by making suitable substitutions in the following functions defined as:(v)  \sqrt{\frac{a^{2}+x^{2}}{a^{2}-x}}
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1. Find \frac{d y}{d x} by making suitable substitutions in the following functions defined as:(v) \sqrt{\frac{a^{2}+x^{2}}{a^{2}-x}}

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