Calculus problem

Question

Determine the slope of the graph of x⁴ = ln(xy) at the point (1, e)

Options:

1. 3e
2. e⁴
3. 1
4. 4

Answer 1

With the help of the rules of logarithms, we can make the equation a little more calculus-friendly, as follows:

x⁴ = ln x + ln y

This will make it easier to do implicit differentiation:

4x³ = ¹/_x + (¹/_y) •(^dy/_dx)

Now we can substitute the given x and y values:

4 • 1³ = ¹/₁ + (¹/e) •(^dy/_dx)

4 = 1 + (¹/e) •(^dy/_dx)

3 = (¹/e) •(^dy/_dx)

3e = ^dy/_dx

So Option 1 is the correct answer.

Answer 2

This problem is best solved using implicit differentiation.

So, to find the slope, just plug in x = 1 and y = e