Use the chain rule to solve: [tex]y=\sqrt{3 x^2+4 x-1}[/tex]
Answer (1)
Rewrite the function as y = ( 3 x 2 + 4 x − 1 ) 2 1 .
Apply the chain rule: d x d y = d u d y ⋅ d x d u , where u = 3 x 2 + 4 x − 1 .
Find d u d y = 2 1 u − 2 1 = 2 u 1 and d x d u = 6 x + 4 .
Multiply and simplify: d x d y = 3 x 2 + 4 x − 1 3 x + 2 .
Explanation
Problem Analysis We are given the function y = 3 x 2 + 4 x − 1 and asked to find its derivative with respect to x using the chain rule.
Understanding the Chain Rule The chain rule states that if we have a composite function y = f ( g ( x )) , then the derivative d x d y is given by d x d y = f ′ ( g ( x )) g ′ ( x ) . In other words, d x d y = d u d y ⋅ d x d u , where u = g ( x ) .
Identifying Inner and Outer Functions Let's rewrite the given function as y = ( 3 x 2 + 4 x − 1 ) 2 1 . We can identify the inner function as u = 3 x 2 + 4 x − 1 and the outer function as y = u 2 1 .
Derivative of the Outer Function Now, we find the derivative of the outer function with respect to u : d u d y = 2 1 u − 2 1 = 2 u 1 .
Derivative of the Inner Function Next, we find the derivative of the inner function with respect to x : d x d u = 6 x + 4 .
Applying the Chain Rule Applying the chain rule, we multiply the derivatives: d x d y = d u d y ⋅ d x d u = 2 u 1 ⋅ ( 6 x + 4 ) = 2 3 x 2 + 4 x − 1 6 x + 4 .
Simplifying the Result Finally, we simplify the expression: d x d y = 2 3 x 2 + 4 x − 1 2 ( 3 x + 2 ) = 3 x 2 + 4 x − 1 3 x + 2 .
Final Answer Therefore, the derivative of y = 3 x 2 + 4 x − 1 with respect to x is: d x d y = 3 x 2 + 4 x − 1 3 x + 2 .
Examples
The chain rule is fundamental in physics, especially when dealing with related rates problems. For instance, consider a scenario where the radius of a circular oil spill is expanding over time. If we know how the area of the spill relates to the radius ( A = π r 2 ) and how the radius changes with time ( d t d r ), we can use the chain rule to find how the area of the spill is changing with time ( d t d A = d r d A ⋅ d t d r = 2 π r d t d r ). This allows environmental scientists to predict the spread of pollutants and implement timely containment strategies.
