The correct answer is **A. 5(2+x)(2x+x²)4.**
To find the derivative of f(x) = (2x+x²)5, we can use the chain rule. The chain rule states that if we have a function of the form f(g(x)), then the derivative of f with respect to x is [1]given by:
$f'(x) = f'(g(x)) * g'(x)$
In this case, let g(x) = 2x+x². Then, f(g(x)) = (2x+x²)5.
To find f'(g(x)), we can use the power rule, which states that if we have a function of the form xn, then the derivative of x with respect to x is given by:
$x^n = nx^{n-1}$
In this case, n = 5, so:
$f'(g(x)) = 5(2x+x²)4$
To find g'(x), we can use the sum rule, which states that if we have two functions f(x) and g(x), then the derivative of f(x) + g(x) with respect to x is given by:
$(f(x) + g(x))' = f'(x) + g'(x)$
In this case, g(x) = 2x+x², so:
$g'(x) = 2 + 2x$
Now, we can substitute f'(g(x)) and g'(x) into the chain rule formula:
$f'(x) = f'(g(x)) * g'(x) = 5(2x+x²)4 * (2 + 2x)$
Simplifying this expression, we get:
$f'(x) = 10(2+x)(2x+x²)4}$
Therefore, the correct answer is **A. 5(2+x)(2x+x²)4.**
