2.2.1 Answer

2.2.1

1.

(a) $${\left(\frac{3x - 1}{x^2 + 1}\right)^{\prime} = \frac{3(x^2 + 1) - (3x-1)(2x)}{(x^2 + 1)^2} = \frac{-3x^2 + 2x + 3}{(x^2 + 1)^2}}$$

(b) $${(\sec{x})^{\prime} = \left(\frac{1}{\cos{x}}\right)^{\prime} = \sec{x}\tan{x}}$$

(c) $${(\csc{x})^{\prime} = \left(\frac{1}{\sin{x}}\right)^{\prime} = - \csc{x}\cot{x}}$$

(d) $${(\cot{x})^{\prime} = \left(\frac{\cos{x}}{\sin{x}}\right)^{\prime} = -\frac{1}{\sin^{2}{x}}}$$

2.

(a) $${(\sin{(x^{2} + 1)})^{\prime} = 2x\cos{(x^2 + 1)}}$$

$${(\cos{\sqrt{x+ 1}})^{\prime} = -\sin{\sqrt{x+1}}(\sqrt{x+1})^{\prime} = -\frac{\sin{(\sqrt{x+1})}}{2\sqrt{x+1}}}$$

(c) $${(e^{\sin{x}})^{\prime} = e^{\sin{x}}(\sin{x})^{\prime} = \cos{x} e^{\sin{x}}}$$

Exercise

1.

Find the derivative of the following function using the derivative of inverse．

(a)

$${y = \cos^{-1}{x}}$$

(b)

$${y = \tan^{-1}{x}}$$

2.

Find the derivative of the following function using the logarithmic differentiation．

(a)

$${y = x^{2}\sqrt{\frac{x^{3} + 2x + 1}{x^{2} - 3x + 1}}}$$

(b)

$${y = x^{x}}$$

(c)

$${y = \sin{x}^{x}}$$

(d)

$${y = x^{1/x}}$$

3.

Find $\frac{dy}{dx}$．

(a)

$${x = a\cos{t}, y = a\sin{t}, \ a > 0}$$

(b)

$${x = \sqrt{t} - \frac{1}{t}, y = t + \frac{1}{\sqrt{t}}}$$

4.

Find the derivative of the following function．

(a)

$${x^{2}(1 + \sqrt{x})}$$

(b)

$${x^{3}\tan{2x}}$$

(c)

$${x\sin^{-1}{x}}$$

(d)

$${\frac{x}{x^{2}+1}}$$

(e)

$${x\sin{x}}$$

(f)

$${x\sin^{-1}x + \sqrt{1-x^{2}}}$$

(g)

$${\tan^{-1}(x^{2} + 1)}$$

(h)

$${\cos{(\sqrt{2x+1})}}$$

(i)

$${\frac{\sin{x} - x\cos{x}}{x\sin{x} + \cos{x}}}$$

(j)

$${e^{2x}\cos{x}}$$

(k)

$${\log{\vert x + \sqrt{x^{2} + A}\vert}}$$