**Example 1 :**

Differentiate the following

cos^{-1}(1 -x^{2})/(1+x^{2})

**Solution :**

Let y = cos^{-1}(1 -x^{2})/(1+x^{2})

x = tan θ

y = cos^{-1}(1- tan^{2} θ)/(1+tan^{2} θ)

y = cos^{-1}(cos 2θ)

y = 2θ

Differentiate with respect to x, we get

dy/dx = 2 (dθ/dx) ------(1)

x = tan θ

1 = sec^{2}θ (dθ/dx)

dθ/dx = 1/sec^{2}θ

By applying the value of dθ/dx in (1), we get

dy/dx = 2 (1/sec^{2}θ)

= 2/(1 + tan^{2}θ)

= 2/(1 + x^{2})

**Example 2 :**

Differentiate the following

sin^{-1}(3x - 4x^{3})

**Solution :**

Let y = sin^{-1}(3x - 4x^{3})

x = sin θ

y = sin^{-1}(3 sin θ - 4(sin θ)^{3})

y = sin^{-1}(3 sin θ - 4 sin^{3} θ)

y = sin^{-1}(sin 3θ)

y = 3θ

Differentiate with respect to x, we get

dy/dx = 3 (dθ/dx) ------(1)

x = sin θ

1 = cos θ (dθ/dx)

dθ/dx = 1/cosθ

By applying the value of dθ/dx in (1), we get

dy/dx = 3 (1/cos sec^{2}θ)

= 2/(1 + tan^{2}θ)

= 2/(1 + x^{2})

**Example 3 :**

Differentiate the following

tan^{-1}[(cos x + sin x) / (cos x - sin x)]

**Solution :**

Let y = tan^{-1}[(cos x + sin x) / (cos x - sin x)]

Divide every terms inside the parentheses by cos x, we get

y = tan^{-1}[(1 + tan x) / (1 - tan x)]

y = tan^{-1}[(tan π/4) + tan x) / (1 - (tan π/4) tan x)]

y = tan^{-1}[tan ((π/4) + x)]

y = (π/4) + x

Differentiate with respect to "x", we get

dy/dx = 1

**Example 4 :**

Find the derivative of sin x^{2} with respect to x^{2}

**Solution :**

Let y = sin x^{2}

here we have to differentiate with respect to "x^{2" }not "x"

dy/dx^{2 }= cosx^{2}

Incase we differentiate with respect to "x", we get

dy/dx = cosx^{2 }(2x)

= 2x cosx^{2}

**Example 5 :**

Find the derivative of sin^{-1}(2x / (1 + x^{2})) with respect to tan^{-1} x

**Solution :**

Let y = sin^{-1}(2x / (1 + x^{2}))

x = tan θ

y = sin^{-1}(2tan θ / (1 + tan^{2} θ))

y = sin^{-1}(2tan θ / (1 + tan^{2} θ))

y = sin^{-1}(sin 2θ)

y = 2θ

y = 2 tan^{-1}x

dy/dx = 2 [1/(1 + x^{2})]

dy/dx = 2/(1 + x^{2})

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit the following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**