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Second Year Math Differentiation


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

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.ApplytheMaclaurinseriesexpansiontoprovethat:(ii)cosx=1x22+x44x616+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

Findthederivativeofxatx=afromfirstprincipleFind 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

Example5:Findthederivativeofx3+2x+3Example 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.Finddydxif(iii)y=tanh1(sinx)π2xπ23. 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)

Example3:Finddydxify=a+x+axa+xax(x0)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)

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