How do you find f'(x) using the definition of a derivative # f(x) = x^2 + x#?

2 Answers
Jul 23, 2018

# 2x+1#.

Explanation:

Recall that, #f'(x)=lim_(t to x){f(t)-f(x)}/(t-x)............(ast)#.

#f(x)=x^2+x rArr f(t)=t^2+t#.

#:. f(t)-f(x)=(t^2-x^2)+(t-x)#,

#=(t-x)(t+x)+(t-x)#,

# rArr f(t)-f(x)=(t-x){(t+x)+1}#.

#:. {f(t)-f(x)}/(t-x)={(t+x)+1}, (t!=x)#.

#:.," by "(ast), "f'(x)=lim_(t to x){(t+x)+1}#,

# i.e., f'(x)={(x+x)+1}=2x+1#, as desired!

#color(blue)("Enjoy Maths.!")#

Jul 23, 2018

#f'(x)=2x+1#

Explanation:

#"differentiating from first principles"#

#f'(x)=lim_(hto0)(f(x+h)-f(x))/h#

#=lim_(hto0)((x+h)^2+x+h-x^2-x)/h#

#=lim_(hto0)(cancel(x^2)+2hx+h^2cancel(+x)+hcancel(-x^2)cancel(-x))/h#

#=lim_(hto0)(cancel(h)(2x+h+1))/cancel(h)#

#=2x+1#