What is (\partial)/(\partial x)(cos^6(x^9y^3)) ?

The answer to (\partial)/(\partial x)(cos^6(x^9y^3)) is -54x^8y^3cos^5(x^9y^3)sin(x^9y^3)

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(\partial)/(\partial x)(cos^6(x^9y^3))

\frac{\partial}{\partial x} (cos^{6} (x^{9} y^{3}))

- 54 x^{8} y^{3} cos^{5} (x^{9} y^{3}) sin (x^{9} y^{3})

Solution steps

\frac{\partial}{\partial x} (cos^{6} (x^{9} y^{3}))

Treat

y

as a constant

Apply the chain rule:

6 (cos (x^{9} y^{3}))^{5} \frac{\partial}{\partial x} (cos (x^{9} y^{3}))

= 6 (cos (x^{9} y^{3}))^{5} \frac{\partial}{\partial x} (cos (x^{9} y^{3}))

\frac{\partial}{\partial x} (cos (x^{9} y^{3})) = - sin (x^{9} y^{3}) \cdot 9 y^{3} x^{8}

= 6 (cos (x^{9} y^{3}))^{5} (- sin (x^{9} y^{3}) \cdot 9 y^{3} x^{8})

Simplify

= - 54 x^{8} y^{3} cos^{5} (x^{9} y^{3}) sin (x^{9} y^{3})

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What is (\partial)/(\partial x)(cos^6(x^9y^3)) ?
The answer to (\partial)/(\partial x)(cos^6(x^9y^3)) is -54x^8y^3cos^5(x^9y^3)sin(x^9y^3)