Use Chain Rule. [tex]$y=\left(3 x^2-1\right)^2$[/tex]
Answer (2)
Identify the outer and inner functions: y = u 2 and u = 3 x 2 − 1 .
Find the derivatives: d u d y = 2 u and d x d u = 6 x .
Apply the chain rule: d x d y = d u d y ⋅ d x d u = 2 u ⋅ 6 x = 2 ( 3 x 2 − 1 ) ⋅ 6 x .
Simplify to find the final derivative: 36 x 3 − 12 x .
Explanation
Problem Analysis We are given the function y = ( 3 x 2 − 1 ) 2 and asked to find its derivative using the chain rule. The chain rule is used when we have a composite function, meaning a function inside another function. In this case, we have the outer function u 2 and the inner function u = 3 x 2 − 1 .
Chain Rule The chain rule states that if we have a composite function y = f ( g ( x )) , then the derivative of y with respect to x is given by d x d y = d u d y ⋅ d x d u , where u = g ( x ) .
Derivative of Outer Function First, let's find the derivative of the outer function with respect to u . If y = u 2 , then d u d y = 2 u .
Derivative of Inner Function Next, let's find the derivative of the inner function with respect to x . If u = 3 x 2 − 1 , then d x d u = 6 x .
Applying the Chain Rule Now, we apply the chain rule: d x d y = d u d y ⋅ d x d u = 2 u ⋅ 6 x . Substitute u = 3 x 2 − 1 back into the equation: d x d y = 2 ( 3 x 2 − 1 ) ⋅ 6 x .
Simplifying the Expression Finally, simplify the expression: d x d y = 12 x ( 3 x 2 − 1 ) = 36 x 3 − 12 x .
Final Answer Therefore, the derivative of y = ( 3 x 2 − 1 ) 2 with respect to x is 36 x 3 − 12 x .
Examples
Consider a scenario where you're analyzing the power output of a solar panel. The power output might depend on the angle of the sun, which changes throughout the day. If the angle is represented by a function g ( x ) and the power output as a function of the angle is f ( g ( x )) , using the chain rule helps you determine how the power output changes with respect to time, d x df = d g df ⋅ d x d g . This is crucial for optimizing the panel's positioning to maximize energy capture.
The derivative of the function y = ( 3 x 2 − 1 ) 2 is calculated using the chain rule, resulting in d x d y = 36 x 3 − 12 x . First, we identify the outer function and the inner function, then find their derivatives and apply the chain rule. Finally, we simplify to arrive at the final answer.
;
