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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}

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:$\text { (xi) } x^{m} m \in N$

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

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}

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)