If [tex]f(x)=\sin \left(x^2\right)[/tex], find [tex]f^{\prime}(x)[/tex]
Answer (2)
Apply the chain rule to f ( x ) = sin ( x 2 ) .
Find the derivative of the outer function: d u d sin ( u ) = cos ( u ) .
Find the derivative of the inner function: d x d x 2 = 2 x .
Combine the results to get the final answer: 2 x cos ( x 2 ) .
Explanation
Problem Analysis We are given the function f ( x ) = sin ( x 2 ) and asked to find its derivative, f ′ ( x ) . This requires applying the chain rule.
Applying the Chain Rule The chain rule states that if we have a composite function f ( g ( x )) , then its derivative is f ′ ( g ( x )) × g ′ ( x ) . In our case, f ( u ) = sin ( u ) and g ( x ) = x 2 .
Derivative of Outer Function First, we find the derivative of the outer function, f ( u ) = sin ( u ) , with respect to u : d u d sin ( u ) = cos ( u )
Derivative of Inner Function Next, we find the derivative of the inner function, g ( x ) = x 2 , with respect to x : d x d x 2 = 2 x
Combining the Results Now, we apply the chain rule: f ′ ( x ) = cos ( g ( x )) × g ′ ( x ) = cos ( x 2 ) × 2 x = 2 x cos ( x 2 )
Final Answer Therefore, the derivative of f ( x ) = sin ( x 2 ) is f ′ ( x ) = 2 x cos ( x 2 ) .
Examples
Consider a scenario where you're analyzing the motion of a pendulum, and its angular displacement is given by sin ( t 2 ) . Finding the rate of change of this displacement (i.e., its derivative) helps you understand the pendulum's angular velocity at any given time. This is crucial in physics for studying oscillatory motion and understanding how variables change with respect to each other.
The derivative of f ( x ) = sin ( x 2 ) is calculated using the chain rule, resulting in f ′ ( x ) = 2 x cos ( x 2 ) . This involves finding the derivatives of the outer function sin and the inner function x 2 and then combining them. Hence, the final answer is 2 x cos ( x 2 ) .
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