# Derivative of $3 e^{- 4 r} \sin{\left(3 \theta \right)}$ with respect to $r$

The calculator will find the derivative of $3 e^{- 4 r} \sin{\left(3 \theta \right)}$ with respect to $r$, with steps shown.

Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps

Leave empty for autodetection.
Leave empty, if you don't need the derivative at a specific point.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Find $\frac{d}{dr} \left(3 e^{- 4 r} \sin{\left(3 \theta \right)}\right)$.

### Solution

Apply the constant multiple rule $\frac{d}{dr} \left(c f{\left(r \right)}\right) = c \frac{d}{dr} \left(f{\left(r \right)}\right)$ with $c = 3 \sin{\left(3 \theta \right)}$ and $f{\left(r \right)} = e^{- 4 r}$:

$${\color{red}\left(\frac{d}{dr} \left(3 e^{- 4 r} \sin{\left(3 \theta \right)}\right)\right)} = {\color{red}\left(3 \sin{\left(3 \theta \right)} \frac{d}{dr} \left(e^{- 4 r}\right)\right)}$$

The function $e^{- 4 r}$ is the composition $f{\left(g{\left(r \right)} \right)}$ of two functions $f{\left(u \right)} = e^{u}$ and $g{\left(r \right)} = - 4 r$.

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

$$3 \sin{\left(3 \theta \right)} {\color{red}\left(\frac{d}{dr} \left(e^{- 4 r}\right)\right)} = 3 \sin{\left(3 \theta \right)} {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dr} \left(- 4 r\right)\right)}$$

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

$$3 \sin{\left(3 \theta \right)} {\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dr} \left(- 4 r\right) = 3 \sin{\left(3 \theta \right)} {\color{red}\left(e^{u}\right)} \frac{d}{dr} \left(- 4 r\right)$$

$$3 e^{{\color{red}\left(u\right)}} \sin{\left(3 \theta \right)} \frac{d}{dr} \left(- 4 r\right) = 3 e^{{\color{red}\left(- 4 r\right)}} \sin{\left(3 \theta \right)} \frac{d}{dr} \left(- 4 r\right)$$

Apply the constant multiple rule $\frac{d}{dr} \left(c f{\left(r \right)}\right) = c \frac{d}{dr} \left(f{\left(r \right)}\right)$ with $c = -4$ and $f{\left(r \right)} = r$:

$$3 e^{- 4 r} \sin{\left(3 \theta \right)} {\color{red}\left(\frac{d}{dr} \left(- 4 r\right)\right)} = 3 e^{- 4 r} \sin{\left(3 \theta \right)} {\color{red}\left(- 4 \frac{d}{dr} \left(r\right)\right)}$$

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

$$- 12 e^{- 4 r} \sin{\left(3 \theta \right)} {\color{red}\left(\frac{d}{dr} \left(r\right)\right)} = - 12 e^{- 4 r} \sin{\left(3 \theta \right)} {\color{red}\left(1\right)}$$

Thus, $\frac{d}{dr} \left(3 e^{- 4 r} \sin{\left(3 \theta \right)}\right) = - 12 e^{- 4 r} \sin{\left(3 \theta \right)}$.

$\frac{d}{dr} \left(3 e^{- 4 r} \sin{\left(3 \theta \right)}\right) = - 12 e^{- 4 r} \sin{\left(3 \theta \right)}$A