How do you find the derivative of f(x)=x^-4 using limit definition ?

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Answer 1 · 2018-05-16T15:39:48

#f'(x)=-4x^(-5)=-4/x^5#

We know that,

#color(red)((1)f'(x)=lim_(t->x)(f(t)-f(x))/(t-x)...to#[ limit definition ]

#color(blue)((2)lim_(x->a)(x^n-a^n)/(x-a)=na^(n-1)#

#f(x)=x^-4=1/x^4=>f(t)=1/t^4#

Using #(1)# we get

#f'(x)=lim_(t->x)(f(t)-f(x))/(t-x)#

#=lim_(t->x)(1/t^4-1/x^4)/(t-x)#

#=lim_(t->x)(x^4-t^4)/(t^4x^4(t-x))#

#=lim_(t->x)(-1)/(t^4x^4)xxlim_(t->x)(t^4-x^4)/(t-x)...toApply(2)#

#=(-1)/(x^4x^4)xx(4x^(4-1))#

#=-1/x^8xx4x^3#

#=-4x^(3-8)#

#=-4x^(-5)#

#=-4/x^5#