Using the limit definition, how do you differentiate #f(x)=x^3−7x+5#?

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Answer 1 · 2018-07-03T20:39:58

#f'(x)=3x^2-7#

Given function: #f(x)=x^3-7x+5#

#f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}#

#=\lim_{h\to 0}\frac{(x+h)^3-7(x+h)+5-(x^3-7x+5)}{h}#

#=\lim_{h\to 0}\frac{(x+h)^3-x^3-7h}{h}#

#=\lim_{h\to 0}\frac{(x+h-x)((x+h)^2+x^2+(x+h)x)-7h}{h}#

#=\lim_{h\to 0}\frac{h((x+h)^2+2x^2+hx)-7h}{h}#

#=\lim_{h\to 0}((x+h)^2+2x^2+hx-7)#

#=((x+0)^2+2x^2+0\cdot x-7)#

#=3x^2-7#