Tag: Derivatives

Find the derivative of log (arccos (x^-1/2))

by RoRi

November 29, 2015

Find the derivative of the function

$f(x) = \log \left( \arccos \frac{1}{\sqrt{x}} \right).$

Using the chain rule and the formulas for the derivative of the logarithm and the arccosine we have,

$\begin{align*} f'(x) &= \left( \frac{1}{\arccos \left( \frac{1}{\sqrt{x}} \right)} \right) \left( \frac{-1}{\sqrt{1 - \left( \frac{1}{\sqrt{x}} \right)^2}} \right) \left(\frac{-1}{2x^{\frac{3}{2}}} \right)\\ &= \frac{\sqrt{x}}{2x^{\frac{3}{2}} \sqrt{x-1} \arccos \frac{1}{\sqrt{x}}} \\ &= \frac{1}{2x \sqrt{x-1} \arccos \frac{1}{\sqrt{x}}}. \end{align*}$

The formula is valid for $x > 1$ .

Find the derivative of (arccos (x²))^-2

by RoRi

November 29, 2015

Find the derivative of the function

$f(x) = \frac{1}{(\arccos(x^2))^2}.$

Here we use the power rule, the chain rule, and the formula for the derivative of the arccosine,

$\begin{align*} f'(x) &= (-2)(\arccos(x^2))^{-3} (2x) \left( \frac{-1}{\sqrt{1-x^4}} \right) \\ &= \frac{4x}{\sqrt{1-x^4} (\arccos(x^2))^3}. \end{align*}$

The formula is valid for $|x| < 1$ .

Find the derivative of arctan ((1+x)/(1-x))

by RoRi

November 29, 2015

Find the derivative of the function

$f(x) = \arctan \frac{1+x}{1-x}.$

Using the chain rule and the formula for the derivative of arctangent we have

$\begin{align*} f'(x) &= \left( \frac{1-x + 1 + x}{(1-x)^2} \right) \left( \frac{1}{1+\left( \frac{1+x}{1-x}\right)^2} \right) \\[9pt] &= \left( \frac{2}{(1-x)^2} \right) \left( \frac{(1-x)^2}{(1-x)^2 + (1+x)^2} \right) \\[9pt] &= \frac{2}{2+2x^2} \\[9pt] &= \frac{1}{1+x^2}. \end{align*}$

This is valid for $x \neq 1$ .

Find the derivative of arccos (1-x²)^1/2

by RoRi

November 29, 2015

Find the derivative of the function

$f(x) = \arccos \sqrt{1-x^2}.$

To compute the derivative we use the chain rule and the formula for the derivative of arccosine,

$\begin{align*} f'(x) &= \left(\frac{-x}{\sqrt{1-x^2}}\right) \left(\frac{-1}{\sqrt{1-1+x^2}} \right) \\ &= \frac{x}{|x| \sqrt{1-x^2}}. \end{align*}$

This formula is valid for $0 < |x| < 1$ .

Find the derivative of arcsin(sin x – cos x)

by RoRi

November 29, 2015

Find the derivative of the function

$f(x) = \arcsin (\sin x - \cos x).$

Using the formula for the derivative of the arcsine and the chain rule we have

$\begin{align*} f'(x) &= (\cos x + \sin x) \frac{1}{\sqrt{1 - (\sin x - \cos x)^2}} \\ &= \frac{\sin x + \cos x}{\sqrt{1 - (\sin^2 x - \sin (2x) + \cos^2 x)}} \\ &= \frac{\sin x + \cos x}{\sqrt{\sin (2x)}}. \end{align*}$

This is valid for all $x$ such that $\sin (2x) \neq 0$ .

Find the derivative of arctan (x+(1+x²)^1/2)

by RoRi

November 29, 2015

Find the derivative of the function

$f(x) = \arctan \left( x + \sqrt{1+x^2} \right).$

Using the formula for the derivative of arctangent and the chain rule we compute this,

$\begin{align*} f'(x) &= \left( 1+ \frac{x}{\sqrt{1+x^2}} \right) \left( \frac{1}{1 + (x+\sqrt{1+x^2})^2} \right) \\ &= \left( \frac{x + \sqrt{1+x^2}}{\sqrt{1+x^2}} \right) \left( \frac{1}{2 + 2x^2 + 2x \sqrt{1+x^2} } \right) \\ &= \frac{1}{2} \left( \frac{x + \sqrt{1+x^2}}{\sqrt{1+x^2}} \right) \left( \frac{1}{(\sqrt{1+x^2})(x + \sqrt{1+x^2})} \right) \\ &= \frac{1}{2(1+x^2)}. \end{align*}$

This is valid for all $x$ .

Find the derivative of arctan (tan² x)

by RoRi

November 29, 2015

Find the derivative of the function

$f(x) = \arctan(\tan^2 x).$

Using the formula for the derivative of the tangent and arctangent we compute,

$\begin{align*} f'(x) &= (2 \tan x \sec^2 x) \left( \frac{1}{1+\tan^4 x} \right) \\[9pt] &= \frac{2 \frac{\sin x}{\cos^3 x} \cos^4 x}{\sin^4 x + \cos^4 x} \\[9pt] &= \frac{2 \sin x \cos x}{\sin^4 x + \cos^4 x} \\[9pt] &= \frac{\sin (2x)}{\sin^4 x + \cos^4 x}. \end{align*}$

This formula is valid everywhere $\tan x$ is defined (i.e., everywhere $\cos x \neq 0$ ).

Find the derivative of arcsin ((1-x²)/(1+x²))

by RoRi

November 29, 2015

Find the derivative of the function

$f(x) = \arcsin \frac{1-x^2}{1+x^2}.$

Using the formula for the derivative of arcsine and the chain rule we have,

$\begin{align*} f'(x) &= \left( \frac{-4x}{(1+x^2)^2} \right) \left( \frac{1}{\sqrt{1- \left( \frac{1-x^2}{1+x^2} \right)^2}} \right) \\[9pt] &= \left( \frac{-4x}{(1+x^2)^2} \right) \left( \frac{1+x^2}{\sqrt{(1+x^2)^2 - (1-x^2)^2}} \right) \\[9pt] &= \frac{-4x}{(1+x^2)\sqrt{4x^2}} \\[9pt] &= - \frac{2x}{|x| (1+x^2)}. \end{align*}$

This is valid for $x \neq 0$ .

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Apostol - Calculus 1

Exercises

Derivatives

Trig Functions

Find the derivative of arctan x + (1/3) arctan (x³)

by RoRi

November 29, 2015

Find the derivative of the function

$f(x) = \arctan x + \frac{1}{3} \arctan (x^3).$

Using the formula for the derivative of arctangent and the chain rule we compute,

$\begin{align*} f'(x) &= \frac{1}{1+x^2} + \frac{1}{3} (3x^2) \left( \frac{1}{1+(x^3)^2} \right) \\[9pt] &= \frac{1}{1+x^2} + \frac{x^2}{1+x^6} \\[9pt] &= \frac{1+x^6+x^2+x^4}{(1+x^2)(1+x^6)} \\[9pt] &= \frac{(1+x^2)(1+x^4)}{(1+x^2)(1+x^6)} \\[9pt] &= \frac{1+x^4}{1+x^6}. \end{align*}$

This is valid for all $x$ .

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Apostol - Calculus 1

Exercises

Derivatives

Trig Functions

Find the derivative of x^1/2 – arctan (x^1/2)

by RoRi

November 29, 2015

Find the derivative of the function

$f(x) = \sqrt{x} - \arctan \sqrt{x}.$

Using the formula for the derivative of arctangent and the chain rule we have

$\begin{align*} f'(x) &= \frac{1}{2\sqrt{x}} - \left( \frac{1}{2\sqrt{x}} \right) \left( \frac{1}{1+(\sqrt{x})^2} \right) \\ &= \left( \frac{1}{2 \sqrt{x}} \right) \left( 1- \frac{1}{1+x} \right) \\ &= \left( \frac{1}{ 2\sqrt{x}} \right) \left( \frac{x}{1+x} \right) \\ &= \frac{\sqrt{x}}{2(1+x)}. \end{align*}$

This is valid for $x \geq 0$ .

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Apostol - Calculus 1

Exercises

Derivatives

Trig Functions

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Tag: Derivatives

Find the derivative of log (arccos (x^-1/2))

Find the derivative of (arccos (x²))^-2

Find the derivative of arctan ((1+x)/(1-x))

Find the derivative of arccos (1-x²)^1/2

Find the derivative of arcsin(sin x – cos x)

Find the derivative of arctan (x+(1+x²)^1/2)

Find the derivative of arctan (tan² x)

Find the derivative of arcsin ((1-x²)/(1+x²))

Find the derivative of arctan x + (1/3) arctan (x³)

Find the derivative of x^1/2 – arctan (x^1/2)