You want to differentiate the given function:

$y = \frac{e^x}{4x^2}$

Step-by-Step Differentiation

This is a quotient of two functions, so we will use the quotient rule. The quotient rule is given by:

$\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}$

Where:

• $u = e^x$
• $v = 4x^2$

1. Differentiate $u$ and $v$:

• $\frac{du}{dx} = \frac{d}{dx}(e^x) = e^x$
• $\frac{dv}{dx} = \frac{d}{dx}(4x^2) = 8x$

2. Apply the Quotient Rule:

$\frac{dy}{dx} = \frac{(4x^2)(e^x) - (e^x)(8x)}{(4x^2)^2}$

3. Simplify the expression:

• The numerator becomes: $(4x^2)(e^x) - (e^x)(8x) = e^x(4x^2 - 8x)$
• The denominator is: $(4x^2)^2 = 16x^4$

Thus, the derivative is:

$\frac{dy}{dx} = \frac{e^x(4x^2 - 8x)}{16x^4}$

4. Further simplify:

We can factor out $4x$ from the numerator:

$\frac{dy}{dx} = \frac{e^x \cdot 4x(x - 2)}{16x^4}$

Cancel the common factors of $4x$:

$\frac{dy}{dx} = \frac{e^x(x - 2)}{4x^3}$

So, the derivative is:

$\boxed{\frac{dy}{dx} = \frac{e^x(x - 2)}{4x^3}}$

Let me know if you would like more details on any of these steps.

Here are some related questions to explore:

1. How do you differentiate a product of two functions (product rule)?
2. What is the chain rule, and how does it differ from the quotient rule?
3. Can you explain how to simplify expressions with exponents in the numerator and denominator?
4. How would the differentiation change if $e^x$ was replaced with a different function?
5. What is the second derivative of the function $y = \frac{e^x}{4x^2}$?

Tip: When using the quotient rule, always simplify as much as possible to avoid handling complex fractions!