Resolved: "Find the derivatives of the given functions. z(x)=ln|sec⁡x+tan⁡x|" - Plainmath

djeljenike

Answered question

2021-05-30

Find the derivatives of the given functions.

z (x) = \ln | \sec x + \tan x |

Answer & Explanation

Ayesha Gomez

Skilled2021-05-31Added 104 answers

z (x) = \ln | \sec x + \tan x |

Differentiate both sides with respect to x

z^{'} (x) = \frac{d}{d x} [\ln | \sec x + \tan x |]

Apply

\frac{d}{d x} [\ln | u |] = \frac{1}{u} \frac{d u}{d x}

z^{'} (x) = \frac{1}{\sec x + \tan x d x} \frac{d}{d x} [\sec x + \tan x]

Therefore,

z^{'} (x) = \frac{1}{\sec x + \tan x} (\sec x \tan x + \sec^{2} x)

Simplify

z^{'} (x) = \frac{\sec x}{\sec x + \tan x} (\tan x + \sec x)

z^{'} (x) = \sec x

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