Using the limit definition, how do you find the derivative of # f(x)=4x^2#?

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Answer 1 · 2016-05-02T00:41:02

#f'(x)=8x#

The limit definition of a derivative states that

#f'(x)=lim_(hrarr0)(f(x+h)-f(x))/h#

Substituting #f(x)=4x^2# into #f'(x)#,

#f'(x)=lim_(hrarr0)(4(x+h)^2-4x^2)/h#

From this point on, you want to expand and simplify.

#f'(x)=lim_(hrarr0)(4(x^2+2xh+h^2)-4x^2)/h#

#f'(x)=lim_(hrarr0)(4x^2+8xh+4h^2-4x^2)/h#

#f'(x)=lim_(hrarr0)(h(8x+4h))/h#

#f'(x)=lim_(hrarr0)8x+4h#

Plugging in #h=0#,

#f'(x)=8x+4(0)#

#color(green)(|bar(ul(color(white)(a/a)color(black)(f'(x)=8x)color(white)(a/a)|)))#