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

1 Answer
May 2, 2016

f'(x)=8x

Explanation:

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)

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