To find the derivative (dy)/(dx) of the given function, we will differentiate each term separately.

To find the derivative (dy)/(dx) of the given function, we will differentiate each term separately. The given function is: y = (5)/(x^3) - 2 (5x) + (3)/([3]x) + sqrt(5x) First, rewrite the terms using exponent notation: y = 5x^-3 - 2 (5x) + 3x^-1/3 + (5x)^1/2 Now, we differentiate each term: Step 1: Differentiate the first term, 5x^-3. Using the power rule (d)/(dx)(cx^n) = cnx^n-1: (d)/(dx)(5x^-3) = 5(-3)x^-3-1 = -15x^-4 = -(15)/(x^4) Step 2: Differentiate the second term, -2 (5x). Using the chain rule for (u), where u=5x and (du)/(dx)=5: (d)/(dx)(-2 (5x)) = -2 · (1)/(5x) · 5 = -2 · (5)/(5x) = -2 · (1)/(x) = -(2)/(x) Step 3: Differentiate the third term, 3x^-1/3. Using the power rule: (d)/(dx)(3x^-1/3) = 3(-(1)/(3))x^-1/3-1 = -1x^-4/3 = -(1)/(x^4/3) Step 4: Differentiate the fourth term, (5x)^1/2. Using the chain rule for u^n, where u=5x and (du)/(dx)=5: (d)/(dx)((5x)^1/2) = (1)/(2)(5x)^1/2-1 · 5 = (1)/(2)(5x)^-1/2 · 5 = (5)/(2sqrt(5x)) We can simplify (5)/(2sqrt(5x)): (5)/(2sqrt(5x)) = (5)/(2sqrt(5)x) = sqrt(5)sqrt(5)2sqrt(5)sqrt(x) = sqrt(5)2sqrt(x) Step 5: Combine all the derivatives. (dy)/(dx) = -(15)/(x^4) - (2)/(x) - (1)/(x^4/3) + sqrt(5)2sqrt(x) The final answer is -(15)/(x^4) - (2)/(x) - (1)/(x^4/3) + sqrt(5)2sqrt(x). Send me the next one 📸