What is (\partial)/(\partial x)(cos^3(x^3y^8)) ?

The answer to (\partial)/(\partial x)(cos^3(x^3y^8)) is -9x^2y^8cos^2(x^3y^8)sin(x^3y^8)

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(\partial)/(\partial x)(cos^3(x^3y^8))

\frac{\partial}{\partial x} (cos^{3} (x^{3} y^{8}))

- 9 x^{2} y^{8} cos^{2} (x^{3} y^{8}) sin (x^{3} y^{8})

Solution steps

\frac{\partial}{\partial x} (cos^{3} (x^{3} y^{8}))

Treat

y

as a constant

Apply the chain rule:

3 (cos (x^{3} y^{8}))^{2} \frac{\partial}{\partial x} (cos (x^{3} y^{8}))

= 3 (cos (x^{3} y^{8}))^{2} \frac{\partial}{\partial x} (cos (x^{3} y^{8}))

\frac{\partial}{\partial x} (cos (x^{3} y^{8})) = - sin (x^{3} y^{8}) \cdot 3 y^{8} x^{2}

= 3 (cos (x^{3} y^{8}))^{2} (- sin (x^{3} y^{8}) \cdot 3 y^{8} x^{2})

Simplify

= - 9 x^{2} y^{8} cos^{2} (x^{3} y^{8}) sin (x^{3} y^{8})

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What is (\partial)/(\partial x)(cos^3(x^3y^8)) ?
The answer to (\partial)/(\partial x)(cos^3(x^3y^8)) is -9x^2y^8cos^2(x^3y^8)sin(x^3y^8)