What is the derivative of y=(sin x)^x?

mriswith170wp

mriswith170wp

Answered question

2023-02-12

What is the derivative of y = ( sin x ) x ?

Answer & Explanation

Amaqumrhund1

Amaqumrhund1

Beginner2023-02-13Added 9 answers

Step 1
Use logarithmic differentiation.
y = ( sin x ) x
ln y = ln ( ( sin x ) x ) = x ln ( sin x ) (Use properties of ln )
Differentiate implicitely: (Use the product rule and the chain ruel)
1 y d y d x = 1 ln ( sin x ) + x [ 1 sin x cos x ]
So, we have:
1 y d y d x = ln ( sin x ) + x cot x
Solve for d y d x by multiplying by y = ( sin x ) x ,
d y d x = ( ln ( sin x ) + x cot x ) ( sin x ) x
l1voyax

l1voyax

Beginner2023-02-14Added 2 answers

Step 1
d d x [ f ( x ) g ( x ) ]
= [ g ( x ) f ( x ) f ( x ) + g ( x ) ln ( f ( x ) ) ] [ f ( x ) g ( x ) ]

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