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Derivative of $$$e^{9 x}$$$
Related calculator: Derivative Calculator
Solution
The function $$$e^{9 x}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = e^{u}$$$ and $$$g{\left(x \right)} = 9 x$$$.
Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(e^{9 x}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(9 x\right)\right)}$$The derivative of the exponential is $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(9 x\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(9 x\right)$$Return to the old variable:
$$e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(9 x\right) = e^{{\color{red}\left(9 x\right)}} \frac{d}{dx} \left(9 x\right)$$Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 9$$$ and $$$f{\left(x \right)} = x$$$:
$$e^{9 x} {\color{red}\left(\frac{d}{dx} \left(9 x\right)\right)} = e^{9 x} {\color{red}\left(9 \frac{d}{dx} \left(x\right)\right)}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$9 e^{9 x} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 9 e^{9 x} {\color{red}\left(1\right)}$$Thus, $$$\frac{d}{dx} \left(e^{9 x}\right) = 9 e^{9 x}$$$.
Answer
$$$\frac{d}{dx} \left(e^{9 x}\right) = 9 e^{9 x}$$$A