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Instantaneous rate of change of $$$f{\left(x \right)} = 5 x^{x}$$$ at $$$x = 3$$$
Your Input
Find the instantaneous rate of change of $$$f{\left(x \right)} = 5 x^{x}$$$ at $$$x = 3$$$.
Solution
The instantaneous rate of change of the function $$$f{\left(x \right)}$$$ at the point $$$x = x_{0}$$$ is the derivative of the function $$$f{\left(x \right)}$$$ evaluated at the point $$$x = x_{0}$$$.
This means that we need to find the derivative of $$$5 x^{x}$$$ and evaluate it at $$$x = 3$$$.
So, find the derivative of the function: $$$\frac{d}{dx} \left(5 x^{x}\right) = 5 x^{x} \left(\ln\left(x\right) + 1\right)$$$ (for steps, see derivative calculator).
Finally, evaluate the derivative at $$$x = 3$$$.
$$$\left(\frac{d}{dx} \left(5 x^{x}\right)\right)|_{\left(x = 3\right)} = \left(5 x^{x} \left(\ln\left(x\right) + 1\right)\right)|_{\left(x = 3\right)} = 135 + 135 \ln\left(3\right)$$$
Therefore, the instantaneous rate of change of $$$f{\left(x \right)} = 5 x^{x}$$$ at $$$x = 3$$$ is $$$135 + 135 \ln\left(3\right)$$$.
Answer
The instantaneous rate of $$$f{\left(x \right)} = 5 x^{x}$$$A at $$$x = 3$$$A is $$$135 + 135 \ln\left(3\right)\approx 283.312658970194808$$$A.