Answer :
Final answer:
The correct answer for the derivative of the function f(x) = 5(10x⁷ - 7x⁵)¹⁰ is f'(x) = 50x⁴(10x⁷ - 7x⁵)⁹(70x⁶ - 35x⁴). This is found using the chain rule and the power rule for differentiation. So correct option is A.
Explanation:
To differentiate the function f(x) = 5(10x⁷ - 7x⁵)¹⁰, we need to use the chain rule which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Let's first find the derivative of the outer function, which in this case is 5u¹⁰ where u is our inner function 10x⁷ - 7x⁵.
Applying the power rule, the derivative of 5u¹⁰ with respect to u would be 50u⁹.
Next, we determine the derivative of the inner function u = 10x⁷ - 7x⁵.
Again, applying the power rule, we find that the derivative with respect to x is 70x⁶ - 35x⁴.
Now, we multiply the derivative of the outer function by the derivative of the inner function to get the final solution to the differentiation of f(x).
The correct derivative is f'(x) = 50(10x⁷ - 7x⁵)⁹(70x⁶ - 35x⁴).
So the correct option is A. f'(x) = 50x⁴(10x⁷ - 7x⁵)⁹(70x⁶ - 35x⁴).
