Final answer:
To differentiate the function, rewrite the cube and square roots as fractional exponents, apply the power rule for both terms and the quotient rule for the second term, and then simplify and combine the derivatives to find the final answer.
Step-by-step explanation:
The student has been asked to differentiate the function that is the cube root of (3x^2) minus one divided by the square root of (5x). To differentiate this function, we need to apply the rules of differentiation for powers and the quotient rule. The cube root of something can be written as a power of one-third, and the square root as a power of one-half.
Let's denote the original function as f(x). First, we rewrite the cube root and square root as fractional exponents:
f(x) = (3x^2)^(1/3) - (1 / (5x)^(1/2))
Next, we apply the power rule for differentiation, which tells us to multiply by the exponent and reduce the exponent by one. For the derivative of the first term, we get:
f'(x) = (1/3)(3x^2)^(-2/3) * 2 * 3x
And for the second term, we use the quotient rule (vdu - udv) / v^2:
f'(x) continues = - ((1/2)(5x)^(-1/2) * 5) / (5x)
After simplifying, we combine the derivatives of both terms to find the final answer.