Compute derivative using the definition of the derivative.
f(x)=1+1/x^2
1+ 1/(x+h)^2 - (1+ 1/(x^2))
------ divide by h and multiply by reciprocal
1/h(x+h)^2 - 1/hx^2
Next I expanded it and got this messy huge number and multiply the bottom of each other after cancelling some terms out
(2h^2x+h^3)/(h^2x^4+2h^3x^3+h^4x^2) That's as far as I got and I'm not quite sure what I did wrong or what to do next, however I checked online and the actual derivative is -2/x^3
f´(x)= lim [ f ( x + ∆h ) - f ( x ) ] / ∆h
∆h-> 0
In this case :
f ( x ) = 1 + 1 / x ^ 2
f ( x + ∆h ) = 1 + 1 / ( x + ∆h ) ^ 2
f ( x + ∆h ) - f ( x ) =
1 + 1 / ( x + ∆h ) ^ 2 - ( 1 + 1 / x ^ 2 ) =
1 + 1 / ( x + ∆h ) ^ 2 - 1 - 1 / x ^ 2 =
1 / ( x + ∆h ) ^ 2 - 1 / x ^ 2 =
x ^ 2 / [ x ^ 2 * ( x + ∆h ) ^ 2 ] - ( x + ∆h ) ^ 2 / [ x ^ 2 * ( x + ∆h ) ^ 2 ] =
[ x ^ 2 - ( x + ∆h ) ^ 2 ] / [ x ^ 2 * ( x + ∆h ) ^ 2 ] =
[ x ^ 2 - ( x ^ 2 + 2 x ∆h + ∆h ^ 2 ) ] / [ x ^ 2 * ( x + ∆h ) ^ 2 ] =
[ x ^ 2 - x ^ 2 - 2 x ∆h - ∆h ^ 2 ] / [ x ^ 2 * ( x + ∆h ) ^ 2 ] =
[ - 2 x ∆h - ∆h ^ 2 ] / [ x ^ 2 * ( x + ∆h ) ^ 2 ] =
∆h [ - 2 x - ∆h ] / [ x ^ 2 * ( x + ∆h ) ^ 2 ]
f ( x + ∆h ) - f ( x ) = ∆h [ - 2 x - ∆h ] / [ x ^ 2 * ( x + ∆h ) ^ 2 ]
Now:
f´(x)= lim [ f ( x + ∆h ) - f ( x ) ] / ∆h
∆h-> 0
f´ (x)= lim { ∆h [ - 2 x - ∆h ] / [ x ^ 2 * ( x + ∆h ) ^ 2 ] } / ∆h
∆h-> 0
f ´ (x)= lim ∆h [ - 2 x - ∆h ] / { ∆h [ x ^ 2 * ( x + ∆h ) ^ 2 ] }
∆h-> 0
f ´ (x)= lim [ - 2 x - ∆h ] / [ x ^ 2 * ( x + ∆h ) ^ 2 ]
∆h-> 0
As ∆h-> 0 then:
- 2 x - ∆h = - 2 x - 0 = - 2 x
( x + ∆h ) ^ 2 = ( x + 0 ) ^ 2 = x ^ 2
so :
f´(x)= lim [ - 2 x - ∆h ] / [ x ^ 2 * ( x + ∆h ) ^ 2 ] =
∆h-> 0
- 2 x / ( x ^ 2 * x ^ 2 ) =
- 2 x / x ^ 4 =
- 2 x / ( x * x ^ 3 ) =
- 2 / x ^ 3
In your homework you can replace ∆h with h
Is all the same.
To find the derivative of the function using the definition of the derivative, you should take the limit as h approaches 0 of the difference quotient:
f'(x) = lim(h→0) [f(x + h) - f(x)] / h
Let's go through the steps again to find out where the error occurred:
Given function: f(x) = 1 + 1/x^2
1. Evaluate f(x + h):
f(x + h) = 1 + 1/(x + h)^2
2. Compute the difference quotient:
[f(x + h) - f(x)] / h
= [1 + 1/(x + h)^2 - (1 + 1/x^2)] / h
3. Simplify the expression:
= [1/(x + h)^2 - 1/x^2] / h
To combine the fractions, find a common denominator:
= [(x^2 - (x + h)^2) / (x^2(x + h)^2)] / h
= [x^2 - (x^2 + 2hx + h^2) / (x^2(x + h)^2)] / h
Now simplify the numerator further:
= [- 2hx - h^2 / (x^2(x + h)^2)] / h
= [- 2hx - h^2] / (x^2(x + h)^2 * h)
At this point, you can divide each term by h to simplify:
= [-2x - h] / (x^2(x + h)^2)
Now, take the limit as h approaches 0:
lim(h→0) [-2x - h] / (x^2(x + h)^2)
= -2x / (x^2(x)^2)
= -2x / x^4
= -2 / x^3
Therefore, the derivative of f(x) = 1 + 1/x^2 using the definition of the derivative is f'(x) = -2 / x^3.
It seems the error occurred during the simplification of the numerator before taking the limit. Remember to divide each term in the numerator by h and then take the limit as h approaches 0 to get the correct answer.