What is (\partial)/(\partial θ)(r(θ)cos(θ)) ?

The answer to (\partial)/(\partial θ)(r(θ)cos(θ)) is r(cos(θ)-θsin(θ))

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(\partial)/(\partial θ)(r(θ)cos(θ))

\frac{\partial}{\partial θ} (r (θ) cos (θ))

r (cos (θ) - θ sin (θ))

Solution steps

\frac{\partial}{\partial θ} (r (θ) cos (θ))

Treat

r

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= r \frac{\partial}{\partial θ} (θ cos (θ))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = θ, g = cos (θ)

= r (\frac{\partial}{\partial θ} (θ) cos (θ) + \frac{\partial}{\partial θ} (cos (θ)) θ)

\frac{\partial}{\partial θ} (θ) = 1

\frac{\partial}{\partial θ} (cos (θ)) = - sin (θ)

= r (1 \cdot cos (θ) + (- sin (θ)) θ)

Simplify

= r (cos (θ) - θ sin (θ))

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What is (\partial)/(\partial θ)(r(θ)cos(θ)) ?
The answer to (\partial)/(\partial θ)(r(θ)cos(θ)) is r(cos(θ)-θsin(θ))