How do you use conjugates to find the derivative of $f(x)=\sqrt{x}$ from the limit definition?

Q: How do you use conjugates to find the derivative of $f(x)=\sqrt{x}$ from the limit definition?

To find the derivative of $f(x)=\sqrt{x}$, substitute into the difference quotient $\frac{\sqrt{x+h}-\sqrt{x}}{h}$. Multiply the numerator and denominator by the conjugate $\sqrt{x+h}+\sqrt{x}$ to rationalize the numerator. This simplifies the expression to $\frac{1}{\sqrt{x+h}+\sqrt{x}}$. Taking the limit as $h \to 0$ yields $\frac{1}{2\sqrt{x}}$ for $x>0$.

To find the derivative of

f(x)=\sqrt{x}

, substitute into the difference quotient

\frac{\sqrt{x+h}-\sqrt{x}}{h}

. Multiply the numerator and denominator by the conjugate

\sqrt{x+h}+\sqrt{x}

to rationalize the numerator. This simplifies the expression to

\frac{1}{\sqrt{x+h}+\sqrt{x}}

. Taking the limit as

h \to 0

yields

\frac{1}{2\sqrt{x}}

for

x>0

Conditions

$f(x)=\sqrt{x}$ .
$x > 0$ for the derivative to be finite.
$h$ is nonzero during the algebraic manipulation.

Reasoning, step by step

Set up the difference quotient: $\frac{\sqrt{x+h}-\sqrt{x}}{h}$ .
Multiply numerator and denominator by the conjugate: $\frac{(\sqrt{x+h}-\sqrt{x})(\sqrt{x+h}+\sqrt{x})}{h(\sqrt{x+h}+\sqrt{x})}$ .
Apply the difference of squares in the numerator: $(x+h) - x = h$ .
Simplify the fraction: $\frac{h}{h(\sqrt{x+h}+\sqrt{x})}$ .
Cancel $h$ (valid for $h \neq 0$ ): $\frac{1}{\sqrt{x+h}+\sqrt{x}}$ .
Evaluate the limit as $h \to 0$ : $\frac{1}{\sqrt{x}+\sqrt{x}} = \frac{1}{2\sqrt{x}}$ .

Example

The script states: 'For $f(x)$ =√x, multiply by the conjugate to cancel h. At $x>0$ the limit yields $1/(2\sqrt{x})$ ; the formula does not give a finite derivative at $x=0$ .'

Common misconceptions

Forgetting to multiply both the numerator and the denominator by the conjugate.
Attempting to cancel $h$ directly from the radical expression without rationalization.
Assuming the derivative exists and is finite at $x=0$ , whereas the limit diverges.

Watch the explanation

YouTubeDefinition of the Derivative

12:00 – 23:31Watch this moment ↗

Connected concepts

Derivatives →

How do you use conjugates to find the derivative of $f(x)=\sqrt{x}$ from the limit definition?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

Explore next

Related questions

How do you use conjugates to find the derivative of f(x)=xf(x)=\sqrt{x}f(x)=x​ from the limit definition?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

Explore next

Related questions

What is jerk in the context of kinematics and higher-order derivatives?+

Why do two upward-curving graphs at the same point x=4x=4x=4 have different positive second derivatives, 10 versus 0.4?+

How do you express the nearby secant slope using an increment hhh?+

What do the two adjacent dx intervals under the curve represent in the visual explanation of the second derivative notation?+

What is the geometric meaning of the second derivative for a twice differentiable function?+

How do you use conjugates to find the derivative of $f(x)=\sqrt{x}$ from the limit definition?