Find the derivative using first principles? : #sqrt(x-5) #
# d/dx sqrt(x-5) = 1/(2sqrt(x-5)) #
We seek:
By First Principles, using the limit definition:
By signing up, you agree to our Terms of Service and Privacy Policy
To find the derivative of the function ( \sqrt{x-5} ) using first principles, we'll use the definition of the derivative:
[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]
For the given function ( \sqrt{x-5} ), let's denote it as ( f(x) ).
So, ( f(x) = \sqrt{x - 5} ).
Now, we'll substitute ( f(x) ) and ( f(x + h) ) into the definition of the derivative:
[ f'(x) = \lim_{h \to 0} \frac{\sqrt{x + h - 5} - \sqrt{x - 5}}{h} ]
Next, we'll rationalize the numerator by multiplying both the numerator and denominator by the conjugate of the numerator:
[ f'(x) = \lim_{h \to 0} \frac{\sqrt{x + h - 5} - \sqrt{x - 5}}{h} \times \frac{\sqrt{x + h - 5} + \sqrt{x - 5}}{\sqrt{x + h - 5} + \sqrt{x - 5}} ]
Simplify the expression:
[ f'(x) = \lim_{h \to 0} \frac{(x + h - 5) - (x - 5)}{h(\sqrt{x + h - 5} + \sqrt{x - 5})} ]
[ f'(x) = \lim_{h \to 0} \frac{x + h - 5 - x + 5}{h(\sqrt{x + h - 5} + \sqrt{x - 5})} ]
[ f'(x) = \lim_{h \to 0} \frac{h}{h(\sqrt{x + h - 5} + \sqrt{x - 5})} ]
[ f'(x) = \lim_{h \to 0} \frac{1}{\sqrt{x + h - 5} + \sqrt{x - 5}} ]
Now, as ( h ) approaches 0, the expression becomes:
[ f'(x) = \frac{1}{2\sqrt{x - 5}} ]
Therefore, the derivative of ( \sqrt{x - 5} ) using first principles is ( \frac{1}{2\sqrt{x - 5}} ).
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.
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 f'(x) using the definition of a derivative for #f(x)=1/sqrt(x)#?
- Using the limit definition, how do you differentiate #f(x)=1-x^2#?
- How do you find the points where the graph of the function #f(x) = x^4-4x+5# has horizontal tangents and what is the equation?
- What is the equation of the tangent to the curve # y=9tanx # at the point where #x=(2pi)/3#?
- What is the equation of the normal line of #f(x)=x^2-x+5# at #x=2#?

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