What is the derivative of 5 to the x?
Understand the Problem
The question is asking for the derivative of the exponential function 5 raised to the power of x. To find this, we apply the rule for differentiating exponential functions.
Answer
$5^x \ln(5)$
Answer for screen readers
The derivative of $5^x$ is $5^x \ln(5)$
Steps to Solve
- Identify the form of the exponential function
The given function is $5^x$, which is an exponential function of the form $a^x$, where $a$ is a constant.
- Recall the differentiation rule for exponential functions
The derivative of an exponential function $a^x$ is given by the formula: $$\frac{d}{dx}(a^x) = a^x , \ln(a)$$
- Apply the differentiation rule
Using the rule, the derivative of $5^x$ is: $$\frac{d}{dx}(5^x) = 5^x , \ln(5)$$
The derivative of $5^x$ is $5^x \ln(5)$
More Information
The natural logarithm function, denoted $\ln$, appears frequently in the differentiation of exponential functions. It represents the logarithm to the base $e$, where $e$ is approximately equal to 2.71828.
Tips
A common mistake when differentiating exponential functions is forgetting to multiply by the natural logarithm of the base. Always remember to include $\ln(a)$ when differentiating $a^x$.
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