Differentiate the function #f(x)= x/(x+1)# from first principle?
2 Answers
Explanation:
#color(blue)"differentiating from first principles"#
#f'(x)=lim_(hto0)(f(x+h)-f(x))/h#
#color(white)(f'(x))=lim_(hto0)((x+h)/(x+h+1)-x/(x+1))/h#
#"the aim now is to eliminate h from the denominator"#
#color(white)(f'(x))=lim_(hto0)((x+h)(x+1)-x(x+h+1))/(h(x+1)(x+h+1)#
#color(white)(f'(x))=lim_(hto0)(cancel(x^2)cancel(+hx)cancel(+x)+hcancel(-x^2)cancel(-hx)cancel(-x))/(h(x+1)(x+h+1))#
#color(white)(f'(x))=lim_(hto0)cancel(h)^1/(cancel(h)^1(x+1)(x+h+1)#
#color(white)(f'(x))=1/(x+1)^2#
# d/dx x/(1+x) =1/(1+x)^2 #
Explanation:
The definition of the derivative of
# f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h #
So if
And so the derivative of
#f'(x) = lim_(h rarr 0) ( (x+h)/(1+(x+h)) - x/(1+x) ) /h #
# \ \ \ \ \ \ \ \ \ = lim_(h rarr 0) 1/h( (x+h)/(1+x+h) - x/(1+x) ) #
# \ \ \ \ \ \ \ \ \ = lim_(h rarr 0) 1/h( (x+h)(1+x)- x(1+x+h)) /((1+x+h)(1+x) ) #
# \ \ \ \ \ \ \ \ \ = lim_(h rarr 0) ( x+h + x^2+hx - x-x^2-hx) /(x(1+x+h)(1+x) ) #
# \ \ \ \ \ \ \ \ \ = lim_(h rarr 0) ( h ) /(h(1+x+h)(1+x) ) #
# \ \ \ \ \ \ \ \ \ = lim_(h rarr 0) ( 1 ) /((1+x+h)(1+x) ) #
# \ \ \ \ \ \ \ \ \ = ( 1 ) /((1+x+0)(1+x) ) #
# \ \ \ \ \ \ \ \ \ = ( 1 ) /((1+x)(1+x) ) #
# \ \ \ \ \ \ \ \ \ = ( 1 ) /(1+x)^2 #