Final answer:
The student's question requires applying implicit differentiation on the equation x⁷y = 3, resulting in dy/dx = -7/y after applying the product rule and solving for dy/dx.
Step-by-step explanation:
The question involves implicitly differentiating the expression x⁷y = 3. When we perform implicit differentiation, we differentiate both sides of the equation with respect to x, while treating y as a function of x (y = f(x)). This means we need to use the product rule on the left side since we have a product of a function of x (x⁷) and a function y (f(x)).
Using the product rule (uv)' = u'v + uv', our differentiation looks like this:
d/dx(x⁷y) = d/dx(3)
7x⁶y + x⁷(dy/dx) = 0
Now we solve for dy/dx:
dy/dx = -7x⁶y / x⁷
dy/dx = -7/y
To check our solution, we can substitute dy/dx back into the original equation and verify if the differentiation is correct.