How do you use the definition of a derivative to find the derivative of #1/x^2#?

Answer 1

Samantha Adkison

# -2/x^3 #

# f'(x) = lim_(h→ 0 )( f(x+h) - f(x))/h #

# = lim_(h→0) ( 1/(x+h)^2 -1/x^2)/h #

# = lim_(h→0)( (x^2 - (x+h)^2)/(x^2(x+h)^2))/h #

# = lim_(h→0)(( x^2 - x^2 - 2hx - h^2)/(x^2(x+h)^2))/h #

# = lim_(h→0) (( -h(2x+ h))/(x^2(x+h)^2))/h#

# = lim_(h→0) 1/cancel(h) (- cancel(h) (2x+h))/(x^2(x+h)^2) #

# =( - 2x)/x^4 = -2/x^3 #

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Answer 2

Paisley Johnson

To find the derivative of (1/x^2) using the definition of a derivative, apply the limit definition of the derivative (f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}) to the function (1/x^2).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.