Question

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

Answer 1

The slope of the graph is 3e

Solution-

The given equation-

`\Rightarrow x^4=\ln(xy)`

Using logarithm properties,

`\Rightarrow x^4=\ln(x)+\ln(y)`

Taking derivatives of both sides,

`\Rightarrow (d)/(dx)(x^4)=(d)/(dx)(\ln x+\ln y)`

`\Rightarrow (d)/(dx)(x^4)=(d)/(dx)(\ln x)+(d)/(dx)(\ln y)`

Applying chain rule,

`\Rightarrow (d)/(dx)(x^4)=(d)/(dx)(\ln x)+(d)/(dy)(\ln y)(dy)/(dx)`

`\Rightarrow 4x^3=(1)/(x)+(1)/(y)(dy)/(dx)`

`\Rightarrow (1)/(y)(dy)/(dx)=4x^3-(1)/(x)`

`\Rightarrow (dy)/(dx)=y(4x^3-(1)/(x))`

Slope of
`x^4=\ln(xy)` at
`(1,e)` is

`=(dy)/(dx)_(at\ (1,e))\\\\=e(4(1)^3-(1)/(1))\\\\=e(4-1)\\\\=3e`