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Nth Derivative <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1"><span class="MJXp-mstyle" id="MJXp-Span-2"><span class="MJXp-mrow" id="MJXp-Span-3"><span class="MJXp-msup" id="MJXp-Span-4"><span class="MJXp-mi" id="MJXp-Span-5" style="margin-right: 0.05em;">Sin</span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-6" style="vertical-align: 0.5em;"><span class="MJXp-mn" id="MJXp-Span-7">4</span></span></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-8">x</span></span></span></span></span><span id="MathJax-Element-1-Frame" class="mjx-chtml MathJax_CHTML MJXc-processing" tabindex="0"></span><script type="math/asciimath" id="MathJax-Element-1">Sin^(4)x</script> | Successive Differentiation | Leibnitz Theorem Solved Example - Math Traders

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Nth Derivative Sin4x | Successive Differentiation | Leibnitz Theorem Solved Example

Nth derivative of Sin^(4)x

In this quick tutorial you will learn that how to find the nth derivative of a trigonometric function like Sin(x) or Sin^4(x) by successive differentiation method. So observe each and every step very carefully and meanwhile if you have any doubt related to the solution at any step please leave us a comment we assure you that you will hear from us as soon as possible. So let’s now…..

 

Given:                                  y =  sin4x   

                             

                              Multiply and divide sin4x by 4 we get


                                             y =  (2sin2x)24


Or                                         y =  (1-cos2x)24            Since (2sin2x)2=(1-cos2x)2 


Now                     expanding (1-cos2x)2 we get the following result


i.e                                      y =  [(1+cos22x-2cos2x)4]


Again,                   multiply and divide the term (cos2x)2 with 4 we get the following result.


i.e.,                                      y =  [(1+4cos22x4-2.cos2x)4]


Or                                         y =  [14+116+116(1+cos4x)2-12cos2x]


Or                                         y =  [14+116+116(1+cos24x+2cos4x)-12cos2x]


Again Multiply & Divide `cos^(2)4x by 2 we get a result which is differentiable easily.


Or          y =  [14+116+116(1+2cos24x2+2cos4x)-12cos2x]


Or          y =  [14+116+116(1+1+cos8x2+2cos4x)-12cos2x]


Or          y =  [14+116+1161+132+cos8x32+cos4x8-12cos2x]



Now                     since nth derivative of sin(ax+b) is:


                                             =  [an.sin(ax+b+n.π2]


So,                        now writing nth derivative of each term we get.


                                            = 1328n.sin(8x+n.π2)  - 184n.sin(4x+n.π2) -  

`\frac{1}{2}{.2^n}.\cos (2x + n.\frac{\pi }{2})`



Nth Derivative `Sin^(4)x`






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