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Calculating the Derivative
We want to calculate the derivative of the function f : ℝ → ℝ with the rule:
f(x) = (8x - 6)3/5.
To find the derivative, we use the chain rule. The chain rule states that if f(x) = (g(x))n, the derivative is given by:
f'(x) = n ⋅ (g(x))n-1 ⋅ g'(x).
Applying this rule to the given function f(x) = (8x - 6)3/5:
- g(x) = 8x - 6
- n = 3/5
- The derivative of g(x) is g'(x) = 8
Now we can calculate the derivative f'(x):
f'(x) = (3/5) (8x - 6)(3/5) - 1 ⋅ 8.
Simplify the exponent:
f'(x) = (3/5) (8x - 6)-2/5 ⋅ 8.
Simplify the constant factors:
f'(x) = (24/5) (8x - 6)-2/5.
So the correct derivative is:
f'(x) = (24/5) (8x - 6)-2/5.
None of the given choices match this exact form, but this is the correct derivative of the function.