Answer :
Final answer:
To differentiate the function (5x⁶+2x³)⁴, we use the chain rule and the power rule. We set u = 5x⁶ + 2x³, then find the derivative of u⁴ with respect to u, and multiply it by the derivative of u with respect to x, giving us the final answer.
Explanation:
The student asked how to differentiate the function (5x⁶+2x³)⁴. To solve this, we use the chain rule, which allows us to differentiate composite functions.
First, let's set u = 5x⁶ + 2x³, which makes our function look like u⁴. The derivative of u⁴ with respect to u would be 4u³.
Now, we need to find the derivative of u with respect to x, du/dx, which is du/dx = (5x⁶)' + (2x³)'. Using the power rule, we know that (xⁿ)' = nxⁿ⁻¹. Considering the constants 5 and 2 as factors that 'come along for the ride', we obtain 30x⁵ and 6x² respectively.
Combining these, we have du/dx = 30x⁵ + 6x². Using the chain rule, the derivative of the original function is then d/dx(u⁴) = 4u³(du/dx) = 4(5x⁶+2x³)³(30x⁵+6x²).
