What is (\partial)/(\partial x)(cos^5(x^6y^7)) ?

The answer to (\partial)/(\partial x)(cos^5(x^6y^7)) is -30x^5y^7cos^4(x^6y^7)sin(x^6y^7)

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(\partial)/(\partial x)(cos^5(x^6y^7))

\frac{\partial}{\partial x} (cos^{5} (x^{6} y^{7}))

- 30 x^{5} y^{7} cos^{4} (x^{6} y^{7}) sin (x^{6} y^{7})

Solution steps

\frac{\partial}{\partial x} (cos^{5} (x^{6} y^{7}))

Treat

y

as a constant

Apply the chain rule:

5 (cos (x^{6} y^{7}))^{4} \frac{\partial}{\partial x} (cos (x^{6} y^{7}))

= 5 (cos (x^{6} y^{7}))^{4} \frac{\partial}{\partial x} (cos (x^{6} y^{7}))

\frac{\partial}{\partial x} (cos (x^{6} y^{7})) = - sin (x^{6} y^{7}) \cdot 6 y^{7} x^{5}

= 5 (cos (x^{6} y^{7}))^{4} (- sin (x^{6} y^{7}) \cdot 6 y^{7} x^{5})

Simplify

= - 30 x^{5} y^{7} cos^{4} (x^{6} y^{7}) sin (x^{6} y^{7})

y = 10 (3 x + 1) (1 - 5 x)

\int arccot (3 y) d y

\int x^{3} x d x

\int_{0}^{1} \frac{12}{4 x ^{2} + 5 x + 1} d x

What is (\partial)/(\partial x)(cos^5(x^6y^7)) ?
The answer to (\partial)/(\partial x)(cos^5(x^6y^7)) is -30x^5y^7cos^4(x^6y^7)sin(x^6y^7)