differentiate from first principles
y=1/x^2
i get to this then i get stuck
f(x+h)-f(x) = 1/(x+h)^2 - 1/x^2
= x^2 - x^2 -2xh -h^2
above divided by(x^2+2xh+h^2)(x^2)
= -2xh -h^2/x^4+2x^3h+h^2x^2
then i know i need to divide by h but i cant seem to get to what i want
Lim x→0 f(x)
should read
f'(x)
To differentiate the function y = 1/x^2 using first principles, you have correctly written the difference quotient:
f(x+h) - f(x) = 1/(x+h)^2 - 1/x^2
Now let's simplify the expression step by step:
f(x+h) - f(x) = (1/(x^2 + 2xh + h^2)) - (1/x^2)
To combine the fractions, we need a common denominator. The common denominator is x^2(x^2 + 2xh + h^2), so we can rewrite the expression as:
f(x+h) - f(x) = (x^2 - (x^2 + 2xh + h^2))/(x^2(x^2 + 2xh + h^2))
Simplifying further:
f(x+h) - f(x) = (x^2 - x^2 - 2xh - h^2)/(x^2(x^2 + 2xh + h^2))
= (-2xh - h^2)/(x^2(x^2 + 2xh + h^2))
Now, as you correctly mentioned, we need to divide the expression by h:
(f(x+h) - f(x))/h = (-2xh - h^2)/(x^2(x^2 + 2xh + h^2)) * (1/h)
Dividing each term by h:
(f(x+h) - f(x))/h = (-2x - h)/(x^2(x^2 + 2xh + h^2))
At this point, you can take the limit of this expression as h approaches zero to find the derivative of y = 1/x^2 using first principles:
lim(h->0) ((f(x+h) - f(x))/h) = lim(h->0) ((-2x - h)/(x^2(x^2 + 2xh + h^2)))
To evaluate this limit, substitute h = 0 into the expression:
lim(h->0) ((-2x - h)/(x^2(x^2 + 2xh + h^2))) = (-2x - 0)/(x^2(x^2 + 2x(0) + 0^2))
= -2x/(x^2(x^2))
= -2/x^3
Therefore, the derivative of y = 1/x^2 using first principles is -2/x^3.
You're on the right track. You just have to expand the numerator, without expanding the denominator.
The reason is because in the numerator, you would like to see the x² terms disappear, surfacing the -2xh term which is critical to cancel the h in the denominator. As h→0, (x+h)² would evaluate like x² in the denominator. The complete works would look like:
f(x)=1/x²
Lim x→0 f(x)
=Lim x→0 (f(x+h)-f(x))/h
=Lim x→0 (1/(x+h)²-1/x²)/h
Lim x→0 (x²-x²-2xh-h²)/(h(x+h)²x²))
=Lim x→0 (-2xh-h²)/(h(x+h)²x²))
=Lim x→0 (-2x-h) / (x+h)²x²)
=-2/x³