- #1
karush
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$\d{}{x}{3}^{x}\ln\left({3}\right)=$
I tried the product rule but didn't get the answer
I tried the product rule but didn't get the answer
karush said:How about the $3^x$
karush said:How about the $3^x$
The Product Rule is a rule in calculus that is used to find the derivative of a product of two functions. It states that the derivative of a product is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function.
In this case, we have a product of two functions: $3^{x}$ and $\ln\left({3}\right)$. Using the Product Rule, we can find the derivative of this product by taking the derivative of the first function, which is $3^{x}$, and multiplying it by the second function, which is $\ln\left({3}\right)$. We then add this to the derivative of the second function, which is $\d{}{x}\ln\left({3}\right)$, multiplied by the first function, $3^{x}$.
The derivative of $3^{x}$ is equal to $3^{x}$ multiplied by the natural logarithm of the base, which in this case is 3. Therefore, the derivative of $3^{x}$ is $3^{x}\ln\left({3}\right)$.
The derivative of $\ln\left({3}\right)$ is equal to $\d{}{x}\ln\left({3}\right) = \frac{1}{x}$.
Yes, the Product Rule can be extended to any number of functions. The general rule for finding the derivative of a product of n functions is $\d{}{x}\left(f_{1}f_{2}...f_{n}\right) = \sum_{i=1}^{n} f_{1}f_{2}...f_{i-1}\d{}{x}f_{i}f_{i+1}...f_{n}$.