Second derivative of $$$e^{5 x}$$$

The calculator will find the second derivative of $$$e^{5 x}$$$, with steps shown.

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Your Input

Find $$$\frac{d^{2}}{dx^{2}} \left(e^{5 x}\right)$$$.

Solution

Find the first derivative $$$\frac{d}{dx} \left(e^{5 x}\right)$$$

The function $$$e^{5 x}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = e^{u}$$$ and $$$g{\left(x \right)} = 5 x$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(e^{5 x}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(5 x\right)\right)}$$

The derivative of the exponential is $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:

$${\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(5 x\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(5 x\right)$$

Return to the old variable:

$$e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(5 x\right) = e^{{\color{red}\left(5 x\right)}} \frac{d}{dx} \left(5 x\right)$$

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 $$$c = 5$$$ and $$$f{\left(x \right)} = x$$$:

$$e^{5 x} {\color{red}\left(\frac{d}{dx} \left(5 x\right)\right)} = e^{5 x} {\color{red}\left(5 \frac{d}{dx} \left(x\right)\right)}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$5 e^{5 x} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 5 e^{5 x} {\color{red}\left(1\right)}$$

Thus, $$$\frac{d}{dx} \left(e^{5 x}\right) = 5 e^{5 x}$$$.

Next, $$$\frac{d^{2}}{dx^{2}} \left(e^{5 x}\right) = \frac{d}{dx} \left(5 e^{5 x}\right)$$$

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 $$$c = 5$$$ and $$$f{\left(x \right)} = e^{5 x}$$$:

$${\color{red}\left(\frac{d}{dx} \left(5 e^{5 x}\right)\right)} = {\color{red}\left(5 \frac{d}{dx} \left(e^{5 x}\right)\right)}$$

The function $$$e^{5 x}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = e^{u}$$$ and $$$g{\left(x \right)} = 5 x$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$$5 {\color{red}\left(\frac{d}{dx} \left(e^{5 x}\right)\right)} = 5 {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(5 x\right)\right)}$$

The derivative of the exponential is $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:

$$5 {\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(5 x\right) = 5 {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(5 x\right)$$

Return to the old variable:

$$5 e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(5 x\right) = 5 e^{{\color{red}\left(5 x\right)}} \frac{d}{dx} \left(5 x\right)$$

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 $$$c = 5$$$ and $$$f{\left(x \right)} = x$$$:

$$5 e^{5 x} {\color{red}\left(\frac{d}{dx} \left(5 x\right)\right)} = 5 e^{5 x} {\color{red}\left(5 \frac{d}{dx} \left(x\right)\right)}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$25 e^{5 x} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 25 e^{5 x} {\color{red}\left(1\right)}$$

Thus, $$$\frac{d}{dx} \left(5 e^{5 x}\right) = 25 e^{5 x}$$$.

Therefore, $$$\frac{d^{2}}{dx^{2}} \left(e^{5 x}\right) = 25 e^{5 x}$$$.

Answer

$$$\frac{d^{2}}{dx^{2}} \left(e^{5 x}\right) = 25 e^{5 x}$$$A