Expert Answer to "first derivative of the given function y=6sec3x"

High School
Algebra
Pre-Algebra
Equations, expressions, and inequalitie

Answered question

2022-01-06

first derivative of the given function

$y = 6 s e c 3 x$

Answer & Explanation

star233

Skilled2022-02-09Added 403 answers

Find the derivative of the following via implicit differentiation:
$\frac{d}{d x} (y) = \frac{d}{d x} (6 \sec (3 x))$

Using the chain rule, $\frac{d}{d x} (y) = \frac{d y (u)}{d u} \frac{d u}{d x}$ , where $u = x$ and $\frac{d}{d u} (y (u)) = y^{'} (u) :$
$(\frac{d}{d x} (x)) y^{'} (x) = \frac{d}{d x} (6 \sec (3 x))$

The derivative of x is 1:
$1 y^{'} (x) = \frac{d}{d x} (6 \sec (3 x))$

Factor out constants:
$y^{'} (x) = 6 (\frac{d}{d x} (\sec (3 x)))$

Using the chain rule, $\frac{d}{d x} (\sec (3 x)) = \frac{d \sec (u)}{d u} \frac{d u}{d x}$ , where $u = 3 x$ and $\frac{d}{d u} (\sec (u)) = \sec (u) \tan (u) :$
$y^{'} (x) = 6 (\frac{d}{d x} (3 x)) \sec (3 x) \tan (3 x)$
INTERMEDIATE STEPS:
Possible derivation:
$\frac{d}{d x} (\sec (x))$
Rewrite the expression: $\sec (x) = \frac{1}{\cos (x)} :$
$= \frac{d}{d x} (\frac{1}{\cos (x)})$
Using the chain rule, $\frac{d}{d x} (\frac{1}{\cos (x)}) = \frac{d}{d u} \frac{1}{u} \frac{d u}{d x}$ , where $u = \cos (x)$ and $\frac{d}{d u} (\frac{1}{u}) = - \frac{1}{u^{2}} :$

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