Answer :
Final answer:
The function f(x) = x⁹e(8x) is differentiated using the product rule, resulting in f'(x) = 9x⁸e(8x) + 72x⁹e(8x), which is option (a).
Explanation:
To differentiate the function f(x) = x⁹e(8x), we need to apply the product rule since the function is a product of two functions: x⁹ and e(8x). According to the product rule (also known as Leibniz's law), if we have a product of two functions u(x) and v(x), then the derivative of their product u(x)v(x) is given by u'(x)v(x) + u(x)v'(x).
In this case, let u(x) = x⁹, so u'(x) = 9x⁸. And let v(x) = e(8x), so v'(x) = 8e(8x) (since the derivative of eⁿⁿ, with respect to n, is itself multiplied by the derivative of the exponent, which in this case is 8).
Applying the product rule:
f'(x) = u'(x)v(x) + u(x)v'(x)
f'(x) = 9x⁸e(8x) + x⁹(8e(8x))
f'(x) = 9x⁸e(8x) + 8x⁹e(8x)
Simplifying, we get:
f'(x) = 9x⁸e(8x) + 72x⁹e(8x)
Therefore, the correct answer is a) f'(x) = 9x⁸e(8x) + 72x⁹e(8x).
