Question

Find the derivative of y with respect to x. y=ln(tan^-1(6x^5))

Answer 1

The derivative of y with respect to x is:

`\[ (dy)/(dx) = (30x^4)/(1 + 36x^(10)) \]`

Step-by-step explanation:

The given function is
`\( y = \ln(\tan^(-1)(6x^5)) \).` To find the derivative
`\( (dy)/(dx) \),` we apply the chain rule. First, we find the derivative of the outer function, which is the natural logarithm
`(\( \ln \)). The derivative of \( \ln(u) \) is \( (1)/(u) \)` times the derivative of
`\( u \). In this case, \( u \) is \( \tan^(-1)(6x^5) \).` Next, we find the derivative of the inner function,
`\( \tan^(-1)(6x^5) \),` using the chain rule again.

The derivative of
`\( \tan^(-1)(v) \) is \( (1)/(1+v^2) \)` times the derivative of
`\( v \). Here, \( v \) is \( 6x^5 \).` Finally, we multiply these two derivatives together. The derivative of the natural logarithm of
`\( \tan^(-1)(6x^5) \)` with respect to
`\( x \)` is given by
`\( (1)/(\tan^(-1)(6x^5)) * (1)/(1 + (6x^5)^2) * (d)/(dx)(6x^5) \)`. Simplifying this expression results in the final answer
`\( (dy)/(dx) = (30x^4)/(1 + 36x^(10)) \).`