I posted one before but this is another one of those problems that i need help with! I don't know how to approach this one at all!
Please help
use the definition of the derivative to find f'(x) when
f(x)= (4/x^2)+(2/5*x)
To find the derivative of the given function using the definition of the derivative, we'll follow these steps:
1. Start with the function f(x) = (4/x^2) + (2/5*x).
2. Recall that the definition of the derivative of a function f(x) at a specific value x is given by the limit:
f'(x) = lim(h->0) [f(x+h) - f(x)] / h
3. Substitute the given function f(x) into the above formula:
f'(x) = lim(h->0) [((4/(x+h)^2) + (2/5*(x+h))) - ((4/x^2) + (2/5*x))] / h
4. Simplify the expression inside the limit using algebraic techniques.
f'(x) = lim(h->0) [(4/(x+h)^2 + 2/5(x+h) - 4/x^2 - 2/5*x)] / h
f'(x) = lim(h->0) [(4/(x^2 + 2hx + h^2) + 2/5(x+h) - 4/x^2 - 2/5*x)] / h
f'(x) = lim(h->0) [(4/(x^2 + 2hx + h^2) - 4/x^2) + (2/5(x+h) - 2/5*x)] / h
5. Next, simplify each term separately.
Term 1: (4/(x^2 + 2hx + h^2) - 4/x^2)
To simplify this term, find a common denominator:
Term 1 = (4x^2 - 4(x^2 + 2hx + h^2)) / (x^2(x^2 + 2hx + h^2))
= (-4hx - 4h^2) / (x^2(x^2 + 2hx + h^2))
Term 2: (2/5(x+h) - 2/5*x)
= [2(x+h) - 2x] / (5(x+h))
= (2h) / (5(x+h))
6. Substitute the simplified terms back into the original expression:
f'(x) = lim(h->0) [(-4hx - 4h^2) / (x^2(x^2 + 2hx + h^2)) + (2h) / (5(x+h))] / h
7. Cancel out the h in the numerator and denominator:
f'(x) = lim(h->0) [-4x - 4h - (4h^2)/(x^2(x^2 + 2hx + h^2)) + (2) / (5(x+h))]
8. Simplify further:
f'(x) = -4x + 2 / (5x)
Therefore, the derivative of f(x) = (4/x^2) + (2/5*x) is f'(x) = -4x + 2 / (5x).