Find the derivative of the given function. $$ y=27^{5 x} $$

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UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$\frac{d}{dx\times 27^{5x}}$$
- step1: Reduce the fraction:
  $$\frac{1}{x\times 27^{5x}}$$
To find the derivative of the function

$$
y = 27^{5x}
$$

we can use the property of exponential functions. The derivative of an exponential function $$a^{u}$$ with respect to $$x$$ is given by:

$$
\frac{d}{dx}(a^{u}) = a^{u} \cdot \ln(a) \cdot \frac{du}{dx}
$$

In our case, $$a = 27$$ and $$u = 5x$$.

1. First, we find $$\frac{du}{dx}$$:
   $$
   \frac{du}{dx} = \frac{d}{dx}(5x) = 5
   $$

2. Now, we can apply the derivative formula:
   $$
   \frac{dy}{dx} = 27^{5x} \cdot \ln(27) \cdot 5
   $$

Thus, the derivative of the function is:

$$
\frac{dy}{dx} = 5 \cdot 27^{5x} \cdot \ln(27)
$$

This is the final result for the derivative of the given function.
The derivative of $$ y = 27^{5x} $$ is $$ \frac{dy}{dx} = 5 \cdot 27^{5x} \cdot \ln(27) $$.

Find the derivative of the given function. \[ y=27^{5 x} \]

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