What is (\partial)/(\partial x)(cos^6(x^9y^5)) ?

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

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(\partial)/(\partial x)(cos^6(x^9y^5))

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

\int \frac{( 2 x + 3 )}{x} d x

y^{^{''}} + 2 y^{^{'}} + 2 y = 8 e^{2 t}, y (0) = 0, y^{^{'}} (0) = 1

\frac{d y}{d t} = 0.5 y, y (2) = 100

\int \frac{x ^{2}}{x ^{3} - 5 x} d x

\int 3^{c s c (x)} csc (x) cot (x) d x

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