# How do you find f'(x) using the definition of a derivative for #f(x)=sqrt(9 − x)#?

Now we have to derivate them:

The only ting left now is to fill in everything we have, into the formula

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

To use the definition see the explanation section below.

Rationalize the numerator.

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) = \sqrt{9 - x} ), you apply the definition of the derivative:

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

Substitute ( f(x) = \sqrt{9 - x} ) into the definition:

[ f'(x) = \lim_{h \to 0} \frac{\sqrt{9 - (x + h)} - \sqrt{9 - x}}{h} ]

Simplify the expression by rationalizing the numerator:

[ f'(x) = \lim_{h \to 0} \frac{\sqrt{9 - x - h} - \sqrt{9 - x}}{h} \cdot \frac{\sqrt{9 - x - h} + \sqrt{9 - x}}{\sqrt{9 - x - h} + \sqrt{9 - x}} ]

[ f'(x) = \lim_{h \to 0} \frac{9 - x - h - (9 - x)}{h(\sqrt{9 - x - h} + \sqrt{9 - x})} ]

[ f'(x) = \lim_{h \to 0} \frac{-h}{h(\sqrt{9 - x - h} + \sqrt{9 - x})} ]

[ f'(x) = \lim_{h \to 0} \frac{-1}{\sqrt{9 - x - h} + \sqrt{9 - x}} ]

Now, let ( h ) approach 0:

[ f'(x) = \frac{-1}{\sqrt{9 - x} + \sqrt{9 - x}} ]

[ f'(x) = \frac{-1}{2\sqrt{9 - x}} ]

Therefore, ( f'(x) = \frac{-1}{2\sqrt{9 - x}} ).

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.

- Using the limit definition, how do you find the derivative of #f(x)=3(x^(-2)) #?
- How do you find the equation of the line tangent to the graph of #f(x)= (ln x)^5# at x=5?
- What is the equation of the line tangent to # f(x)=(x^2-x)e^(x-2) # at # x=-2 #?
- What is the equation of the tangent line of #f(x)=sinx/(x^2-3x+2) # at #x=0#?
- For #f(x) =-x^-3+x^-2#, what is the equation of the line tangent to #x =2 #?

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