Differentiate.
\[ y = \ln(17 - x) \]
Answer (3)
We have to use the chain rule's
f ( x ) = l n ( 17 − x )
f [ g ( x )] = l n [ g ( x )]
therefore
f ( u ) = l n ( u )
and
u = g ( x ) = 17 − x
them we have
f ′ ( x ) = f ′ ( u ) ∗ g ′ ( x )
f ′ ( u ) = u 1
g ′ ( x ) = − 1
f ′ ( x ) = f ′ ( u ) ∗ g ′ ( x )
f ′ ( x ) = u 1 ∗ ( − 1 )
f ′ ( x ) = − u 1
∴ f ′ ( x ) = − 17 − x 1
0\ \ \ \Rightarrow\ \ \ x<17\ \ \ \Rightarrow\ \ \ D=(17;+\infty)"> y ′ = ( 17 − x ) ′ ⋅ l n ( 17 − x ) 1 = − l n ( 17 − x ) 1 an d D : 17 − x > 0 ⇒ x < 17 ⇒ D = ( 17 ; + ∞ )
The derivative of y = ln ( 17 − x ) is d x d y = − 17 − x 1 . This result is obtained using the chain rule by first finding the derivative of the natural logarithm and then the inner function. By applying the chain rule correctly, we arrive at the final answer efficiently.
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