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Derivative of $$$r \cos{\left(\theta \right)}$$$ with respect to $$$r$$$

The calculator will find the derivative of $$$r \cos{\left(\theta \right)}$$$ with respect to $$$r$$$, with steps shown.

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

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Find $$$\frac{d}{dr} \left(r \cos{\left(\theta \right)}\right)$$$.

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 $$$c = \cos{\left(\theta \right)}$$$ and $$$f{\left(r \right)} = r$$$:

$${\color{red}\left(\frac{d}{dr} \left(r \cos{\left(\theta \right)}\right)\right)} = {\color{red}\left(\cos{\left(\theta \right)} \frac{d}{dr} \left(r\right)\right)}$$

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

$$\cos{\left(\theta \right)} {\color{red}\left(\frac{d}{dr} \left(r\right)\right)} = \cos{\left(\theta \right)} {\color{red}\left(1\right)}$$

Thus, $$$\frac{d}{dr} \left(r \cos{\left(\theta \right)}\right) = \cos{\left(\theta \right)}$$$.

Answer

$$$\frac{d}{dr} \left(r \cos{\left(\theta \right)}\right) = \cos{\left(\theta \right)}$$$A

Related Notes: Definition of Derivative , Derivatives of Elementary Functions , Table of Derivatives , Related Rates , Studying Derivative Graphically , Optimization Problems , Applications to Economics , Constant Multiple Rule , Sum and Difference Rules , Product Rule , Quotient Rule , Chain Rule , Derivative of Inverse Function