Find the derivative of $y = \frac{(\log x)^{1/2}}{e^{x^2}}$
Understand the Problem
The question involves finding the derivative of the function y with respect to x, where $y = \frac{(\log x)^{1/2}}{e^{x^2}}$. We'll need to apply calculus rules such as the quotient rule and chain rule.
Answer
$$ \frac{1 - 4x^2 \log x}{2xe^{x^2}\sqrt{\log x}} $$
Answer for screen readers
$$ \frac{dy}{dx} = \frac{1 - 4x^2 \log x}{2xe^{x^2}\sqrt{\log x}} $$
Steps to Solve
- Apply the Quotient Rule
The quotient rule states that if $y = \frac{u}{v}$, then $\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$. Here, $u = (\log x)^{1/2}$ and $v = e^{x^2}$.
- Differentiate u with respect to x
$u = (\log x)^{1/2}$. Let's use the chain rule: $\frac{du}{dx} = \frac{1}{2}(\log x)^{-1/2} \cdot \frac{1}{x} = \frac{1}{2x\sqrt{\log x}}$.
- Differentiate v with respect to x
$v = e^{x^2}$. Again, using the chain rule: $\frac{dv}{dx} = e^{x^2} \cdot 2x = 2xe^{x^2}$.
- Substitute into the Quotient Rule formula
Now we substitute $\frac{du}{dx}$ and $\frac{dv}{dx}$ into the quotient rule formula: $$ \frac{dy}{dx} = \frac{e^{x^2} \cdot \frac{1}{2x\sqrt{\log x}} - (\log x)^{1/2} \cdot 2xe^{x^2}}{(e^{x^2})^2} $$
- Simplify the expression
We can simplify by factoring out $e^{x^2}$ from the numerator and canceling one $e^{x^2}$ from the numerator and denominator: $$ \frac{dy}{dx} = \frac{e^{x^2} \left( \frac{1}{2x\sqrt{\log x}} - 2x\sqrt{\log x} \right)}{e^{2x^2}} = \frac{\frac{1}{2x\sqrt{\log x}} - 2x\sqrt{\log x}}{e^{x^2}}$$ To further simplify, we can combine the terms in the numerator: $$ \frac{dy}{dx} = \frac{\frac{1 - 4x^2 \log x}{2x\sqrt{\log x}}}{e^{x^2}} = \frac{1 - 4x^2 \log x}{2xe^{x^2}\sqrt{\log x}} $$
$$ \frac{dy}{dx} = \frac{1 - 4x^2 \log x}{2xe^{x^2}\sqrt{\log x}} $$
More Information
The derivative represents the instantaneous rate of change of the function $y$ with respect to $x$. It tells us how much $y$ changes for a tiny change in $x$.
Tips
- Forgetting the chain rule: A common mistake is to forget to apply the chain rule when differentiating $(\log x)^{1/2}$ or $e^{x^2}$.
- Incorrectly applying the quotient rule: Ensure the terms in the numerator of the quotient rule are in the correct order and with the correct signs. It's easy to mix up $u$ and $v$ or forget the minus sign.
- Algebraic errors in simplification: Be careful when simplifying the expression after applying the quotient rule. Ensure you are correctly factoring and combining terms.
- Incorrectly differentiating $\log x$: Remember that the derivative of $\log x$ is $\frac{1}{x}$.
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