Using the Product Rule to Solve $\d{}{x}{3}^{x}\ln\left({3}\right)$

In summary, the derivative of $3^x$ is $3^x\ln{(3)}$, as proven using the logarithmic differentiation method. The product rule is not needed because $\ln{(3)}$ is a constant.
  • #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😖
 
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  • #2
Hi karush,

You do not need product rule. $\ln(3)$ is a constant.
 
  • #3
How about the $3^x$
 
  • #4
karush said:
How about the $3^x$

$$3^x = e^{x\ln 3}$$
 
  • #5
karush said:
How about the $3^x$

$\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \, \left( a^x \right) = a^x\,\ln{(a)} \end{align*}$

Proof:

$\displaystyle \begin{align*} y &= a^x \\ \ln{(y)} &= \ln{ \left( a^x \right) } \\ \ln{(y)} &= x\ln{(a)} \\ \frac{\mathrm{d}}{\mathrm{d}x} \, \left[ \ln{(y)} \right] &= \frac{\mathrm{d}}{\mathrm{d}x} \, \left[ x\ln{(a)} \right] \\ \frac{\mathrm{d}}{\mathrm{d}y} \, \left[ \ln{(y)} \right] \, \frac{\mathrm{d}}{\mathrm{d}x} \, \left( y \right) &= \ln{(a)} \\ \frac{1}{y}\,\frac{\mathrm{d}y}{\mathrm{d}x} &= \ln{(a)} \\ \frac{\mathrm{d}y}{\mathrm{d}x} &= y\ln{(a)} \\ \frac{\mathrm{d}y}{\mathrm{d}x} &= a^x\,\ln{(a)} \end{align*}$

Q.E.D.
 

FAQ: Using the Product Rule to Solve $\d{}{x}{3}^{x}\ln\left({3}\right)$

What is the Product Rule?

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.

How is the Product Rule used to solve $\d{}{x}{3}^{x}\ln\left({3}\right)$?

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, $3^{x}$.

What is the derivative of $3^{x}$?

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 $3^{x}\ln\left({3}\right)$.

What is the derivative of $\ln\left({3}\right)$?

The derivative of $\ln\left({3}\right)$ is equal to $\d{}{x}\ln\left({3}\right) = \frac{1}{x}$.

Can the Product Rule be applied to more than two functions?

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 $\d{}{x}\left(f_{1}f_{2}...f_{n}\right) = \sum_{i=1}^{n} f_{1}f_{2}...f_{i-1}\d{}{x}f_{i}f_{i+1}...f_{n}$.

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