Given that [tex]y = \sin \left(2 x^3-4 x+2\right)[/tex], find the derivative of y with respect to x.
A. [tex]2 x^3-4 x \sin \left(2 x^3-4 x+2\right)[/tex]
B. [tex]6 x^2-4 \cos \left(2 x^3-4 x+2\right)[/tex]
C. [tex]4 x^2-4 \sin \left(2 x^3-4 x+2\right)[/tex]
D. [tex] \sin \left(2 x^3-4 x+2\right)[/tex]
Answer (2)
Apply the chain rule to find the derivative of y = sin ( 2 x 3 − 4 x + 2 ) .
Let u = 2 x 3 − 4 x + 2 , so y = sin ( u ) .
Find d u d y = cos ( u ) and d x d u = 6 x 2 − 4 .
Calculate d x d y = d u d y ⋅ d x d u = ( 6 x 2 − 4 ) cos ( 2 x 3 − 4 x + 2 ) . The answer is 6 x 2 − 4 cos ( 2 x 3 − 4 x + 2 ) .
Explanation
Problem Analysis We are given the function y = s in ( 2 x 3 − 4 x + 2 ) and we need to find its derivative with respect to x . This requires the chain rule.
Applying the Chain Rule Let u = 2 x 3 − 4 x + 2 . Then y = s in ( u ) . We have f r a c d y d u = cos ( u ) and f r a c d u d x = 6 x 2 − 4 .
Calculating the Derivative By the chain rule, f r a c d y d x = f r a c d y d u c d o t f r a c d u d x = cos ( u ) c d o t ( 6 x 2 − 4 ) = ( 6 x 2 − 4 ) cos ( 2 x 3 − 4 x + 2 ) .
Final Answer Therefore, the derivative of y with respect to x is ( 6 x 2 − 4 ) cos ( 2 x 3 − 4 x + 2 ) . Comparing this to the given options, we see that option (b) matches our result.
Examples
In physics, if x represents time and y represents the displacement of an object undergoing oscillatory motion, the derivative d x d y gives the object's velocity at any given time. Understanding derivatives helps in analyzing the motion and predicting future positions.
The derivative of y = sin ( 2 x 3 − 4 x + 2 ) with respect to x is ( 6 x 2 − 4 ) cos ( 2 x 3 − 4 x + 2 ) . Therefore, the correct answer is option B: 6 x 2 − 4 cos ( 2 x 3 − 4 x + 2 ) .
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