What is (\partial)/(\partial x)(cos^4(x^3y^5)) ?

The answer to (\partial)/(\partial x)(cos^4(x^3y^5)) is -12x^2y^5cos^3(x^3y^5)sin(x^3y^5)

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(\partial)/(\partial x)(cos^4(x^3y^5))

\frac{\partial}{\partial x} (cos^{4} (x^{3} y^{5}))

- 12 x^{2} y^{5} cos^{3} (x^{3} y^{5}) sin (x^{3} y^{5})

Solution steps

\frac{\partial}{\partial x} (cos^{4} (x^{3} y^{5}))

Treat

y

as a constant

Apply the chain rule:

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

= 4 (cos (x^{3} y^{5}))^{3} \frac{\partial}{\partial x} (cos (x^{3} y^{5}))

\frac{\partial}{\partial x} (cos (x^{3} y^{5})) = - sin (x^{3} y^{5}) \cdot 3 y^{5} x^{2}

= 4 (cos (x^{3} y^{5}))^{3} (- sin (x^{3} y^{5}) \cdot 3 y^{5} x^{2})

Simplify

= - 12 x^{2} y^{5} cos^{3} (x^{3} y^{5}) sin (x^{3} y^{5})

x \to - 8 lim (x^{2} - 3)

f (x) = 4 x^{3} - 8 x

a e^{b t}

f (x) = \frac{3}{x + 2}, at x = 3

sin (9 x) + cos (2 x)

What is (\partial)/(\partial x)(cos^4(x^3y^5)) ?
The answer to (\partial)/(\partial x)(cos^4(x^3y^5)) is -12x^2y^5cos^3(x^3y^5)sin(x^3y^5)