# How do you find f'(x) using the definition of a derivative #f(x)= 1/x#?

Use limit definition of derivative to find:

#f'(x) = -1/x^2#

By signing up, you agree to our Terms of Service and Privacy Policy

To find (f'(x)) using the definition of a derivative for (f(x) = \frac{1}{x}), you use the definition of the derivative:

[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}]

Substitute (f(x) = \frac{1}{x}) into the definition:

[f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h}]

Now, simplify the expression:

[f'(x) = \lim_{h \to 0} \frac{x - (x+h)}{hx(x+h)}]

[f'(x) = \lim_{h \to 0} \frac{-h}{hx(x+h)}]

[f'(x) = \lim_{h \to 0} \frac{-1}{x(x+h)}]

[f'(x) = -\frac{1}{x^2}]

So, the derivative of (f(x) = \frac{1}{x}) with respect to (x) is (f'(x) = -\frac{1}{x^2}).

By signing up, you agree to our Terms of Service and Privacy Policy

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.

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.

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.

- How do you find the instantaneous rate of change for #h(t)=-5t^2+20t+1# for t=2?
- How do you find f'(x) using the limit definition given #f(x)=x^(-1/2)#?
- A particle's acceleration along a straight line is given by #a(t)=48t^2+2t+6# . It's initial velocity is equal to -3cm/s and its initial position is 1 cm. Find its position function s(t). Answer is #s(t)=4t^4+1/3t^3+3t^2-3t+1# but I can't it figure out?
- What is the average rate of change of the function #f(x) = 2x^2 - 3x - 4# for #-3<=x<=-1#?
- What is the equation of the line tangent to the graph of #f(x)= x^4 + 2x^2# at the point where f ' (x)= 1?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7