Get the solution to "Find the derivatives of the functions defined as..."

aramutselv

Answered question

2021-12-07

Find the derivatives of the functions defined as follows :

y = \ln | \tan 2 x |

Navreaiw

Beginner2021-12-08Added 34 answers

Step 1
Given: The function

y = \ln | \tan 2 x |

To determine: The derivative of given function.
Step 2
Explanation:
Differentiating the given function,

\frac{d y}{d x} = \frac{d}{d x} (\ln | \tan 2 x |)

\Rightarrow \frac{d y}{d x} = \frac{d}{d (\tan 2 x)} (\ln | \tan 2 x |) \frac{d}{d (2 x)} (\tan 2 x) \frac{d}{d x} (2 x)

[by chain rule]

\Rightarrow \frac{d y}{d x} = \frac{1}{\tan 2 x} \frac{d}{d (2 x)} (\tan 2 x) \frac{d}{d x} (2 x)

[since

\frac{d}{d x} \ln | x | = \frac{1}{x}

]

\Rightarrow \frac{d y}{d x} = \frac{1}{\tan 2 x} {(\sec 2 x)}^{2} \frac{d}{d x} (2 x)

[since

\frac{d}{d x} \tan x = \sec^{2} x

]

\Rightarrow \frac{d y}{d x} = \frac{1}{\tan 2 x} {(\sec 2 x)}^{2} (2)

[since

\frac{d}{d x} x^{n} = n x^{n - 1}

]

\Rightarrow \frac{d y}{d x} = 2 \frac{\cos 2 x}{\sin 2 x} {(\sec 2 x)}^{2}

\Rightarrow \frac{d y}{d x} = 2 \cos e c 2 x (\sec 2 x)

Answer:

\frac{d y}{d x} = 2 \cos e x 2

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