Final answer:
To find the derivative of the function (ln(x) + 5x + 7)^13, apply the chain rule by first defining u as ln(x) + 5x + 7, then differentiate u^13 and find du/dx separately, finally combine to get the derivative dy/dx.
Step-by-step explanation:
The student is asking about how to evaluate the derivative of a given function raised to a power. The specific function in this case is (ln(x) + 5x + 7)13. To find the derivative, dy/dx, of this function, we apply the chain rule.
Firstly, denote u = ln(x) + 5x + 7. So, the function becomes u13. When differentiating u13 with respect to x, we get:
dy/dx = d/dx [u13] = 13u12 • (du/dx)
Now we need to find du/dx:
du/dx = d/dx [ln(x) + 5x + 7] = 1/x + 5
Therefore, by substituting du/dx into the previous expression, we get:
dy/dx = 13(ln(x) + 5x + 7)12 • (1/x + 5)