What is the derivative of uv?
Understand the Problem
The question is asking for the derivative of the product of two functions, denoted as 'u' and 'v'. This can be solved using the product rule in calculus, which states that the derivative of a product of two functions is given by f'(x)g(x) + f(x)g'(x).
Answer
The derivative of \( u(x) \cdot v(x) \) is \( u'(x) \cdot v(x) + u(x) \cdot v'(x) \).
Answer for screen readers
The derivative of the product ( u(x) \cdot v(x) ) is:
$$ u'(x) \cdot v(x) + u(x) \cdot v'(x) $$
Steps to Solve
- Identify the functions and their derivatives
Let the functions be ( u(x) ) and ( v(x) ), where we need to find the derivatives ( u'(x) ) and ( v'(x) ) respectively.
- Apply the product rule
According to the product rule, the derivative of the product of two functions ( u(x) ) and ( v(x) ) is given by:
$$ \frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) $$
- Substitute the derivatives
After applying the product rule, substitute the values of ( u(x) ), ( v(x) ), ( u'(x) ), and ( v'(x) ) into the formula to obtain the final expression for the derivative of the product.
The derivative of the product ( u(x) \cdot v(x) ) is:
$$ u'(x) \cdot v(x) + u(x) \cdot v'(x) $$
More Information
This equation uses the product rule from calculus, which is essential for finding the derivative of products of functions. It is a fundamental concept in differentiation.
Tips
- Forgetting to apply the product rule correctly (e.g., omitting either part of the sum).
- Not simplifying the resulting expression if it's possible.
- Confusing the order of functions while applying the product rule.
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