Answer :
Final answer:
The derivative of 5e(3x^3-9x^5) can be found by using the chain rule. The derivative is 15x^2e(3x^3 - 9x^5) - 45x^4e(3x^3 - 9x^5), which corresponds to option A. Thus, correct option is (a) 15x²e(3x³ - 9x⁵) - 45x⁴e(3x³ - 9x⁵) .
Explanation:
When using the chain rule to find the derivative of 5e(3x3-9x5), we treat 5e(3x3-9x5) as the product of two functions: the constant 5 and the exponential function e(3x3-9x5). The constant remains unaffected, so we need to focus on differentiating the exponential function.
The derivative of eu, where u is a function of x, is euu'. Applying this rule to our function where u = 3x3 - 9x5, we first find u' by differentiating u with respect to x:
u = 3x3 - 9x5
u' = 9x2 - 45x4
Now, we multiply the derivative of u by the original function to obtain the complete derivative:
(5e(3x3 - 9x5))' = 5e(3x3 - 9x5)(9x2 - 45x4)
By distributing the 5, we get:
5e(3x3 - 9x5)(9x2 - 45x4) = 45x2e(3x3 - 9x5) - 225x4e(3x3 - 9x5
However, we must simplify our answer to match the formats provided in the question. Factoring out e(3x3 - 9x5) we get:
15x2e(3x3 - 9x5) - 45x4e(3x3 - 9x5)
Which corresponds to option A: 15x2e(3x3 - 9x5) - 45x4e(3x3 - 9x5).
