Partial differentiation with respect to x.
Understand the Problem
The question involves calculating partial derivatives of a function with respect to a variable, specifically applying rules of differentiation. It appears to follow a series of steps in deriving a second derivative with respect to x.
Answer
The second partial derivative is given by $$ \frac{\partial^2 u}{\partial x^2} = \frac{f''(x) \cdot x + f'(x)}{y} $$
Answer for screen readers
The final expression for the second partial derivative is: $$ \frac{\partial^2 u}{\partial x^2} = \frac{f''(x) \cdot x + f'(x)}{y} $$
Steps to Solve
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Identify the Function and Variables We are given a function ( f(x) ) and need to calculate the second partial derivative of a variable ( u ) with respect to ( x ).
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First Partial Derivative Calculation Start with the equation for the first derivative: $$ \frac{\partial u}{\partial x} = \frac{f'(x) \cdot x}{y} $$ This expression is derived using the quotient rule.
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Second Partial Derivative Setup To find the second partial derivative, we'll differentiate the first partial derivative again with respect to ( x ): $$ \frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x} \left( \frac{f'(x) \cdot x}{y} \right) $$ Apply the product rule here.
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Applying the Product Rule Using the product rule for differentiation, $$ \frac{\partial^2 u}{\partial x^2} = \frac{1}{y} \left( f''(x) \cdot x + f'(x) \cdot \frac{\partial x}{\partial x} \right) $$ Since ( \frac{\partial x}{\partial x} = 1 ), we have: $$ \frac{\partial^2 u}{\partial x^2} = \frac{1}{y} \left( f''(x) \cdot x + f'(x) \right) $$
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Further Differentiation Continue differentiating to find the required terms from the second partial derivative with respect to ( x ): $$ \frac{\partial^2 u}{\partial x^2} = \frac{1}{y} \left( f''(x) \cdot x + f'(x) \right) $$
The final expression for the second partial derivative is: $$ \frac{\partial^2 u}{\partial x^2} = \frac{f''(x) \cdot x + f'(x)}{y} $$
More Information
This expression describes how the function ( u ) changes with respect to ( x ) and indicates the relationship between the derivatives of ( f(x) ) and the variable ( y ). Understanding partial derivatives is vital in fields such as physics and engineering, where functions depend on multiple variables.
Tips
- Misapplying the Quotient Rule: Ensure the quotient rule is applied correctly for derivatives involving fractions.
- Neglecting the Product Rule: Remember to apply the product rule correctly when dealing with products of functions.
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