\( d \) the derivative of the function \( q(x)=8\left(x^{3}-7 x\right)^{\frac{2}{3}}(x+5)^{\frac{5}{4}} \)

To find the derivative \( d \) of the function \( q(x) = 8(x^{3}-7x)^{\frac{2}{3}}(x+5)^{\frac{5}{4}} \), we will use the product rule and the chain rule. 1. **Product Rule:** If \( u(x) = (x^{3}-7x)^{\frac{2}{3}} \) and \( v(x) = (x+5)^{\frac{5}{4}} \), we have: \[ q'(x) = 8\left(u'(x)v(x) + u(x)v'(x)\right) \] 2. **Finding \( u'(x) \):** \[ u(x) = (x^3 - 7x)^{\frac{2}{3}} \] Applying the chain rule: \[ u'(x) = \frac{2}{3}(x^3 - 7x)^{-\frac{1}{3}}\cdot(3x^2 - 7) \] 3. **Finding \( v'(x) \):** \[ v(x) = (x+5)^{\frac{5}{4}} \] Again, using the chain rule: \[ v'(x) = \frac{5}{4}(x+5)^{\frac{1}{4}} \] 4. **Combining the derivatives:** \[ q'(x) = 8\left(\left(\frac{2}{3}(x^3 - 7x)^{-\frac{1}{3}} \cdot (3x^2 - 7)\right)(x + 5)^{\frac{5}{4}} + (x^3 - 7x)^{\frac{2}{3}} \cdot \left(\frac{5}{4}(x+5)^{\frac{1}{4}}\right)\right) \] Putting everything together, we have: \[ q'(x) = \frac{16}{3}(x^3 - 7x)^{-\frac{1}{3}}(3x^2 - 7)(x + 5)^{\frac{5}{4}} + 10(x^3 - 7x)^{\frac{2}{3}}(x + 5)^{\frac{1}{4}} \] This gives us the derivative \( d \) of the function \( q(x) \).