Final answer:
The derivative of the function f(x) = -15√[5](8x + 5) is found by rewriting the function using exponents, applying the chain rule, and simplifying. The derivative, f'(x), is -24(8x + 5)-4/5.
Step-by-step explanation:
The student has asked to find the derivative of the function f(x) = -15√[5](8x + 5). This function represents a composite function where a constant is multiplied by the fifth root of a binomial expression.
To begin, we rewrite the function using the power rule for roots and exponents. The fifth root of something is the same as raising it to the power of 1/5:
f(x) = -15(8x + 5)1/5
To find the derivative, f'(x), we will use the chain rule. The chain rule dictates that the derivative of a composite function (u(v(x))) is u'(v(x)) multiplied by v'(x). In this case, u(x) = x1/5 and v(x) = 8x + 5.
We find the derivative of u(x) first:
u'(x) = ⅔x-4/5
Next, we find the derivative of v(x):
v'(x) = 8
Now we apply the chain rule:
f'(x) = -15 × ⅔(8x + 5)-4/5 × 8
Simplifying, we get:
f'(x) = -15 × ⅔ × 8 (8x + 5)-4/5
Which simplifies further to:
f'(x) = -24(8x + 5)-4/5
Therefore, the derivative of f(x) = -15√[5](8x + 5) is f'(x) = -24(8x + 5)-4/5.