z = (9x - 4y)^4. Find the derivative.
Answer (2)
Find the partial derivative with respect to x: ∂ x ∂ z = 36 ( 9 x − 4 y ) 3 .
Find the partial derivative with respect to y: ∂ y ∂ z = − 16 ( 9 x − 4 y ) 3 .
Calculate the total derivative: d z = ∂ x ∂ z d x + ∂ y ∂ z d y = 36 ( 9 x − 4 y ) 3 d x − 16 ( 9 x − 4 y ) 3 d y .
The derivatives are ∂ x ∂ z = 36 ( 9 x − 4 y ) 3 , ∂ y ∂ z = − 16 ( 9 x − 4 y ) 3 , and d z = 36 ( 9 x − 4 y ) 3 d x − 16 ( 9 x − 4 y ) 3 d y .
Explanation
Problem Analysis We are given the function z = ( 9 x − 4 y ) 4 and asked to find its derivative. Since the problem does not specify whether we should find the partial derivative with respect to x or y , or the total derivative, I will calculate all three.
Partial Derivative with Respect to x First, let's find the partial derivative of z with respect to x , denoted as ∂ x ∂ z . We will use the chain rule. The chain rule states that if we have a composite function z = f ( g ( x )) , then d x d z = f ′ ( g ( x )) ". g ′ ( x ) . In our case, f ( u ) = u 4 and g ( x ) = 9 x − 4 y . Thus, f ′ ( u ) = 4 u 3 and g ′ ( x ) = 9 . Applying the chain rule, we get: ∂ x ∂ z = 4 ( 9 x − 4 y ) 3 ⋅ ∂ x ∂ ( 9 x − 4 y ) = 4 ( 9 x − 4 y ) 3 ⋅ 9 = 36 ( 9 x − 4 y ) 3
Partial Derivative with Respect to y Next, let's find the partial derivative of z with respect to y , denoted as ∂ y ∂ z . We again use the chain rule. In this case, f ( u ) = u 4 and g ( y ) = 9 x − 4 y . Thus, f ′ ( u ) = 4 u 3 and g ′ ( y ) = − 4 . Applying the chain rule, we get: ∂ y ∂ z = 4 ( 9 x − 4 y ) 3 ⋅ ∂ y ∂ ( 9 x − 4 y ) = 4 ( 9 x − 4 y ) 3 ⋅ ( − 4 ) = − 16 ( 9 x − 4 y ) 3
Total Derivative Finally, let's find the total derivative d z . The total derivative is given by the formula: d z = ∂ x ∂ z d x + ∂ y ∂ z d y Substituting the partial derivatives we found earlier, we get: d z = 36 ( 9 x − 4 y ) 3 d x − 16 ( 9 x − 4 y ) 3 d y
Final Answer The partial derivative of z with respect to x is 36 ( 9 x − 4 y ) 3 , the partial derivative of z with respect to y is − 16 ( 9 x − 4 y ) 3 , and the total derivative is d z = 36 ( 9 x − 4 y ) 3 d x − 16 ( 9 x − 4 y ) 3 d y .
Examples
Understanding derivatives is crucial in physics, especially when analyzing motion. For instance, if z represents the position of an object at time t , the derivative d t d z gives the object's velocity. Similarly, in economics, if z represents a cost function depending on variables x and y , partial derivatives ∂ x ∂ z and ∂ y ∂ z help determine how costs change with respect to changes in x and y , aiding in optimization problems.
We used the chain rule to find the partial derivatives of z = ( 9 x − 4 y ) 4 . The partial derivative with respect to x is 36 ( 9 x − 4 y ) 3 , and the partial derivative with respect to y is − 16 ( 9 x − 4 y ) 3 . The total derivative is d z = 36 ( 9 x − 4 y ) 3 d x − 16 ( 9 x − 4 y ) 3 d y .
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