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Calculus Derivative Example: Product Rule with Logarithms
In calculus, the derivative of a function represents its instantaneous rate of change. This concept is fundamental in understanding the behavior of functions and their applications in various fields. One of the essential tools for finding derivatives is the product rule, which helps us differentiate the product of two functions.
The Product Rule
The product rule states that the derivative of the product of two functions, u(x) and v(x), is equal to the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function. Mathematically, this is represented as:
d/dx [u(x) * v(x)] = u'(x) * v(x) + u(x) * v'(x)
Example: Product Rule with Logarithms
Let’s consider an example to illustrate the application of the product rule with logarithmic functions. Suppose we have the following function:
f(x) = x^2 * ln(x)
Here, u(x) = x^2 and v(x) = ln(x). To find the derivative f'(x), we can apply the product rule:
f'(x) = u'(x) * v(x) + u(x) * v'(x)
First, we need to find the derivatives of u(x) and v(x):
u'(x) = d/dx (x^2) = 2x
v'(x) = d/dx (ln(x)) = 1/x
Now, we can substitute these values back into the product rule equation:
f'(x) = (2x) * ln(x) + x^2 * (1/x)
Simplifying the expression, we get:
f'(x) = 2x * ln(x) + x
Conclusion
The product rule is a fundamental tool in calculus for finding the derivatives of functions involving products. This example demonstrates how to apply the product rule when dealing with logarithmic and exponential functions. By understanding the product rule and its application, students can successfully solve problems involving derivatives in calculus and related fields.
Remember, practice is key to mastering calculus concepts. Try working through additional examples and exercises to solidify your understanding of the product rule and its applications.