Question

Determine the slope of the graph of x^4=ln(xy) at the point (1,e)

Answer 1

It is 3e. Differentiating x^4=ln(xy), we have to implicitly differentiate this.


`4x^(3)= (1)/(xy) \cdot (d)/(dx)(xy)`

The derivative of xy you have to use the chain rule.


`(d)/(dx)(xy)=x \cdot (dy)/(dx)+y`

So now it's


`4x^(3)=(y+(dy)/(dx) \cdot x)/(xy)`

Now we want the slope when x=1 and y=e


`4(1)^(3)=(e+(dy)/(dx))/(e)`


`4=(e+(dy)/(dx))/(e)`


`4=1+(dy)/(dx) \cdot (1)/(e)`


`(dy)/(dx)=3e`

Answer 2

Answer is 3e.

Explanation:

The slope is given by the derivative at the given point.

Differentiating:-

4x^3 = 1/xy *(x * dy/dx + y)

4x^3 = 1/y * dy/dx + 1/x

dy/dx = (4x^3 - 1/x) * y

At the point (1, e) x = 1 and y = e so substituting

slope at (1,e) = dy/dx = (4(1)^3 - 1/1) * e

= 3e (answer)