What is the derivative of cos(x/3 sin(x)) ?

The derivative of cos(x/3 sin(x)) is -1/3 sin(x/3)sin(x)+cos(x)cos(x/3)

What is the first derivative of cos(x/3 sin(x)) ?

The first derivative of cos(x/3 sin(x)) is -1/3 sin(x/3)sin(x)+cos(x)cos(x/3)

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\frac{d}{d x} (cos (\frac{x}{3}) sin (x))

- \frac{1}{3} sin (\frac{x}{3}) sin (x) + cos (x) cos (\frac{x}{3})

Solution steps

\frac{d}{d x} (cos (\frac{x}{3}) sin (x))

Apply the Product Rule:

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

f = cos (\frac{x}{3}), g = sin (x)

= \frac{d}{d x} (cos (\frac{x}{3})) sin (x) + \frac{d}{d x} (sin (x)) cos (\frac{x}{3})

\frac{d}{d x} (cos (\frac{x}{3})) = - sin (\frac{x}{3}) \frac{1}{3}

\frac{d}{d x} (sin (x)) = cos (x)

= (- sin (\frac{x}{3}) \frac{1}{3}) sin (x) + cos (x) cos (\frac{x}{3})

Simplify

= - \frac{1}{3} sin (\frac{x}{3}) sin (x) + cos (x) cos (\frac{x}{3})

What is the derivative of cos(x/3 sin(x)) ?
The derivative of cos(x/3 sin(x)) is -1/3 sin(x/3)sin(x)+cos(x)cos(x/3)
What is the first derivative of cos(x/3 sin(x)) ?
The first derivative of cos(x/3 sin(x)) is -1/3 sin(x/3)sin(x)+cos(x)cos(x/3)