differentiation form first principles y=1/x^2
form or from first principles?
The derivative of y = x^-2 is
dy/dx = -2 x^-3
For a 'first principles' derivation, you will have to use the definition of the derivative and calculate some limits.
I recommend that you perform that exercise yourself.
dy/dx = lim (1/d)[1/(x+d)^2 - 1/d^2]
.......d -> 0
i get so far then i get stuck i think im doing something wrong
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
To find the derivative of the function y = 1/x^2 using the first principles of differentiation, we need to apply the definition of the derivative. The first principles state that the derivative of a function f(x) at a particular point x = a is given by the limit:
f'(a) = lim(h -> 0) [f(a + h) - f(a)] / h
In our case, we want to find the derivative of y = 1/x^2, so we need to evaluate the above limit. Let's go through the steps:
1. Start by substituting the given function into the formula for the derivative:
f'(a) = lim(h -> 0) [1/(a + h)^2 - 1/a^2] / h
2. Simplify the expression inside the limit:
f'(a) = lim(h -> 0) [1/(a^2 + 2ah + h^2) - 1/a^2] / h
3. Combine the fractions by finding a common denominator:
f'(a) = lim(h -> 0) [a^2 - (a^2 + 2ah + h^2)] / [a^2(a^2 + 2ah + h^2)] / h
4. Expand and simplify the numerator:
f'(a) = lim(h -> 0) -[2ah + h^2] / [a^2(a^2 + 2ah + h^2)] / h
5. Factor out h from the numerator:
f'(a) = lim(h -> 0) -[h(2a + h)] / [a^2(a^2 + 2ah + h^2)] / h
6. Simplify by canceling out the common factor of h:
f'(a) = lim(h -> 0) -(2a + h) / [a^2(a^2 + 2ah + h^2)]
7. Now, take the limit as h approaches 0:
f'(a) = -(2a) / [a^2(a^2)] = -2a / a^4
8. Simplify further:
f'(a) = -2 / a^3
Therefore, the derivative of y = 1/x^2 using the first principles of differentiation is f'(x) = -2 / x^3.