# How do you find f'(x) using the definition of a derivative #f(x) =(x-6)^(2/3)#?

Please see the explanation section below.

To save some space, let's do the algebra first, then find the limit.

So, we have

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

Use

This notation can get hard to follow so I am going to make the following substitution:

The numerator becomes what we want:

Now we work with the numerator so that we can make a factor of h cancel out the h in the denominator.

Subtract equation [3] from equation [2]:

Substitute the right side of equation [4] into the numerator of equation [1]:

The h in the numerator cancels the h in the denominator:

Substitute f(x) for b:

Remove a factor of 2 from the numerator:

Remove a factor of 3 from the denominator:

This is what we expected.

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) = (x-6)^{2/3} ), you need to apply the limit definition of the derivative:

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

[ = \lim_{h \to 0} \frac{((x+h)-6)^{2/3} - (x-6)^{2/3}}{h} ]

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.

- What is the equation of the line normal to #f(x)= sqrt(3x^3-2x) # at #x=2#?
- How do you find the equation of a line tangent to the function #y=x^3+6# at (1,7)?
- How do you find the tangent line of #f(x) = 3x/sinx^2 # at x=5?
- How do you find the derivative of #2/sqrtx# using the limit definition?
- How do you find the average rate of change of #y=4x-3x^2# over [2,3]?

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