## DIFFERENTIATION OF RAISE TO POWER FUNCTION

Derivative of lnx, derivative of log x

^{²}, derivative of e^{y},derivative of 2Ë£, Differentiation of raise to power function an easy and short cut manners, differentiation of raise to power function.Till Date we have learn to differentiate this question by taking log on both sides and then differentiate.

There is very long process to differentiate this types of functions .

But today we shall learn a different and an easiest method to differentiate such type of functions.

then assume this function as [ log (Base) ]

**×**[ Power ] then use product rule of differentiation and place the given function in front of the result so obtained.##
Question : Differentiate f(x) = (cos x )^{sin x}

∴ h(x) = (log cos x)

**×**(sin x)

then it derivative will be

f '(x) = h'(x)

Therefore f '(x) = f(x) [(log cos x) . cos x + sin x (- sin x ) /cos x)]

Therefore f '(x) = (cos x )

^{sin x}[(log cos x) . cos x - sin x .tan x]##
Question : How to solve this f(x) = x ^{sin x}* ** *

{ log x . sin x }= ( log x)( sin x ) + ( sin x )( log x)

= log x . cos x + sin x . (1/x)

Now put f(x) in front of this result and that will be derivative of the f(x).

Hence f ' (x) =

*x*{ log x . cos x + sin x . (1/x) }

^{sin x}## Question : Differentiate w.r.t. 'x'

**f(x) = cos x**

^{sin x}+ (sin x)^{x}

*Let*f(x) = g(x) + h(x)

Then f '(x) =g'(x) + h'(x)

Just place

**cos x**in front of derivative of {(log cos x) . (sin x) } + place (

^{sin x}**sin x)**in front of derivative of { ( log sin x) . ( x) },

^{x}So Answer will be

Similarly derivative of h(x) = (sin x)

^{x}in one step can also be written as

h '(x) = (sin x)

^{x}[ log sin x × 1+ x . cos x/sin x ]

##
Question* : *Differentiate f(x) =* **e *^{sin x}

^{sin x}

then using short cut method,

f '(x) =

f '(x) =

*e*[ log e × cos x ] ,^{sin x}f '(x) =

*cos x .**e*^{sin x}## Because derivative of log e is zero and log e is equal to one

##
Question : Differentiate f (x) = *a *^{sin x}

^{sin x}

If f (x) =

*a*^{sin x}
then using short cut method ,

**Question : Differentiate f (x) =**

*x*^{sin x + cos x}

^{sin x + cos x}
then using short cut method ,

f '(x) = x

*[ log x . {cos x - sin x} + {sin x+cos x }.{1/x} ]*^{sin x + cos x}##
**One more shortcut for differentiation you can use**

## Differentiation of Trigonometric Functions

S N f(x)
f '(x)
1
sin x
cos x
2
cos x
-sin x
3
tan x
sec²x
4
cot x
-cosec²x
5
sec x
sec x tan x
6
**cosec x** **-cosec x cot x**

S N | f(x) | f '(x) |
---|---|---|

1 | sin x | cos x |

2 | cos x | -sin x |

3 | tan x | sec²x |

4 | cot x | -cosec²x |

5 | sec x | sec x tan x |

6 | cosec x | -cosec x cot x |

## Conclusion

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