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

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

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Answer 1 · 2016-12-02T16:47:16

$\frac{d y}{d x} = \frac{7}{x \sqrt{x^{14} - 1}}$ $dy/dx = 7/(xsqrt(x^14-1))$

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 $y = {sec}^{- 1} (x^{7}) \Leftrightarrow sec y = x^{7}$ $y=sec^-1(x^7) <=> secy=x^7$

Differentiate Implicitly:

$sec y tan y \frac{d y}{d x} = 7 x^{6}$ $secytanydy/dx = 7x^6$
$:. x^7tanydy/dx = 7x^6$
$:. tanydy/dx = 7/x$ $(x!=0)$ .... [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]
$:. sqrt(x^14-1)dy/dx = 7/x$
$:. dy/dx = 7/(xsqrt(x^14-1))$

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

1 Answer

Explanation:

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What is the derivative of this function y=sec^-1(x^7)y=sec−1(x7)y=sec^-1(x^7)?

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