Final Answer:
The derivative of y = (e^(7x²) * (8x³ + 5)⁴) / ((19x² + 7x - 18)⁵) with respect to x is:
y' = (140e^(7x²) * (8x³ + 5)⁴ * (19x² + 7x - 18)⁴ - 256e^(7x²) * (8x³ + 5) * (19x² + 7x - 18)⁵) / ((19x² + 7x - 18)⁶)
Step-by-step explanation:
To find the derivative of this function, we can use the quotient rule. However, the expression inside the logarithm is quite complicated, making it difficult to apply the quotient rule directly. To simplify the expression, we can take the natural log of both sides:
ln(y) = ln(e^(7x²)) + ln((8x³ + 5)⁴) - ln((19x² + 7x - 18)⁵)
Now, we can apply the chain rule and product rule to find the derivative of this new expression:
y' = (7e^(7x²) * (8x³ + 5)⁴ * (19x² + 7x - 18)) / ((19x² + 7x - 18)⁵) - (4(8x³ + 5)^3 * (19x² + 7x - 18)) / ((19x² + 7x - 18)^5) + (5(8x³ + 5)^4 * (19x² + 7x - 18)^4 * (-38)) / ((19x² + 7x - 18)^6)
Simplifying this expression, we get:
y' = (140e^(7x²) * (8x³ + 5)⁴ * (19x² + 7x - 18)⁴ - 256e^(7x²) * (8x³ + 5) * (19x² + 7x - 18)⁵) / ((19x² + 7x - 18)⁶)
This is our final answer. Note that this expression is still quite complicated, but it is now in a form that can be easily differentiated further if needed.