How do you find f'(x) using the definition of a derivative for #f(x)=1/sqrt(x)#?
The crucial step uses
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}{\sqrt{x}} ), follow these steps:

Start with the definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x + h)  f(x)}{h} ]

Substitute the function ( f(x) = \frac{1}{\sqrt{x}} ) into the formula: [ f'(x) = \lim_{h \to 0} \frac{\frac{1}{\sqrt{x + h}}  \frac{1}{\sqrt{x}}}{h} ]

Rationalize the numerator: [ f'(x) = \lim_{h \to 0} \frac{\sqrt{x}  \sqrt{x + h}}{h \sqrt{x} \sqrt{x + h}} ]

Combine the fractions in the numerator: [ f'(x) = \lim_{h \to 0} \frac{\sqrt{x}  \sqrt{x + h}}{h} \times \frac{1}{\sqrt{x} \sqrt{x + h}} ]

Multiply and divide by the conjugate of the numerator: [ f'(x) = \lim_{h \to 0} \frac{\sqrt{x}  \sqrt{x + h}}{h} \times \frac{\sqrt{x} + \sqrt{x + h}}{\sqrt{x} + \sqrt{x + h}} ]

Simplify the numerator: [ f'(x) = \lim_{h \to 0} \frac{x  (x + h)}{h(\sqrt{x} + \sqrt{x + h})} ]

Cancel out ( x ) terms in the numerator: [ f'(x) = \lim_{h \to 0} \frac{h}{h(\sqrt{x} + \sqrt{x + h})} ]

Cancel out ( h ) terms in the numerator and denominator: [ f'(x) = \lim_{h \to 0} \frac{1}{\sqrt{x} + \sqrt{x + h}} ]

Substitute ( h = 0 ) into the expression: [ f'(x) = \frac{1}{2\sqrt{x} \cdot 1} ]

Simplify the expression: [ \boxed{f'(x) = \frac{1}{2x\sqrt{x}}} ]
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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 onesided 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 onesided 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 onesided 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 equation of the tangent line to the curve # f(x) = ln(x)# at x = 4?
 What is the equation of the line that is normal to #f(x)= xe^(2x2) # at # x= 1 #?
 When can instantaneous rate of change not be calculated?
 A factory produces bicycles at a rate of 80+0.5t^20.7t bicycles per week (t in weeks). How many bicycles were produced from day 15 to 28?
 How do you find f'(x) using the definition of a derivative #f(x)=6 x + 2 /sqrt{x}#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7