Final answer:
To find (d²ʸ/dx²) (d²ˣ/dy²), we differentiate y = e³ˣ with respect to x using the chain rule and differentiate (d²ˣ/dy²) = (d²/dy²)(3ˣ), which results in 0. Multiplying the two derivatives gives us the final answer of 0.
Step-by-step explanation:
To find (d²ʸ/dx²) (d²ˣ/dy²), we first need to differentiate y = e³ˣ with respect to x.
Using the chain rule, we have:
(d/dx)(e³ˣ) = 3e³ˣ * (d/dx)(3ˣ)
Since (d/dx)(3ˣ) = 3ˣ * ln(3), we can substitute it back into the previous equation to get:
(d/dx)(e³ˣ) = 3e³ˣ * 3ˣ * ln(3)
Next, we differentiate (d²ˣ/dy²) = (d²/dy²)(3ˣ).
Since 3ˣ is a constant with respect to y, the second derivative is zero. Therefore, (d²ˣ/dy²) = 0.
Finally, we multiply the two derivatives:
(d²ʸ/dx²) (d²ˣ/dy²) = (3e³ˣ * 3ˣ * ln(3)) * 0 = 0.