Answer :
The derivative of the function f(x) = [tex](5x^2 + 5x + 2)^3[/tex] with respect to x is [tex]60x(5x^2 + 5x + 2)^2.[/tex]
To find the derivative of the given function, we'll apply the chain rule. The chain rule states that if we have a composition of functions, such as [tex](g(h(x)))^n[/tex], the derivative is[tex]n * g'(h(x)) * h'(x)[/tex], where g is the outer function and h is the inner function.
In this case, the outer function is raising to the power of 3, and the inner function is [tex]5x^2 + 5x + 2.[/tex] The derivative of the outer function with respect to the inner function is [tex]3 * (5x^2 + 5x + 2)^2[/tex]. The derivative of the inner function with respect to x is 10x + 5.
Now, applying the chain rule formula:
Derivative of f(x) = [tex]3 * (5x^2 + 5x + 2)^2 * (10x + 5)[/tex]
Simplify:
Derivative of f(x) = [tex]60x(5x^2 + 5x + 2)^2[/tex]
So, the derivative of the function is [tex]60x(5x^2 + 5x + 2)^2.[/tex]
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