Question

What is the derivative of this function `y=sec^-1(x^7)`?

Answer 1

`dy/dx = 7/(xsqrt(x^14-1))`

Explanation:

When tackling the derivative of inverse trig functions. I prefer to rearrange and use Implicit differentiation as I always get the inverse derivatives muddled up, and this way I do not need to remember the inverse derivatives. If you can remember the inverse derivatives then you use the chain rule.

Let `y=sec^-1(x^7) <=> secy=x^7`

Differentiate Implicitly:

`secytanydy/dx = 7x^6`
`:. x^7tanydy/dx = 7x^6`
`:. tanydy/dx = 7/x` `(x!=0)` .... [1]

Using the `sec"/"tan` identity;

`tan^2y+1=sec^2y`
`:. tan^2y+1=(x^7)^2`
`:. tan^2y=x^14-1`
`:. tany=sqrt(x^14-1)`

Substituting into [1]
`:. sqrt(x^14-1)dy/dx = 7/x`
`:. dy/dx = 7/(xsqrt(x^14-1))`