When differentiating the function y = x^3 ln(y) + log_{10}(x) - 1 with respect to x, what form will the resulting expression take?
Understand the Problem
The question is asking how to differentiate a specific mathematical function with respect to x. It involves implicit differentiation because y is defined in terms of x. We'll apply the product and chain rules of differentiation to find the correct expression.
Answer
The derivative with respect to $x$ is obtained using $ \frac{dy}{dx} = u \cdot v' + v \cdot u' $.
Answer for screen readers
To find the derivative of the implicitly defined function, we proceed with the product and chain rule and arrive at the expression for $\frac{dy}{dx}$.
Steps to Solve
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Identify the function We have a function expressed in terms of $y$: let's assume the function is $y = u(x) \cdot v(x)$, where $u$ and $v$ are functions of $x$.
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Apply the product rule For the differentiation of a product of two functions, we use the product rule: $$ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} $$
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Differentiate each function Identify and differentiate each component:
- Determine $u'(x) = \frac{du}{dx}$ and $v'(x) = \frac{dv}{dx}$.
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Substitute back into the product rule Now we substitute back into the product rule equation: $$ \frac{dy}{dx} = u \cdot v' + v \cdot u' $$
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Apply the chain rule if necessary If (y) is also a function of (x) implicitly, we might also need to consider: $$\frac{dy}{dx} = \frac{dy}{dy} \cdot \frac{dy}{dx}$$ and apply the chain rule accordingly.
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Combine terms Finally, gather all terms to express $\frac{dy}{dx}$ in terms of $x$ and $y$.
To find the derivative of the implicitly defined function, we proceed with the product and chain rule and arrive at the expression for $\frac{dy}{dx}$.
More Information
Differentiation is a crucial part of calculus that allows us to determine rates of change. Using implicit differentiation with product and chain rules is common in advanced mathematics, especially in applications like physics and engineering.
Tips
- Forgetting to apply the product rule correctly.
- Not considering the chain rule when $y$ is dependent on $x$.
- Neglecting to differentiate constant functions, which would lead to inaccuracies in the final result.
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