What is (\partial)/(\partial x)(4xy-10x^2y^2z+5z^2) ?

The answer to (\partial)/(\partial x)(4xy-10x^2y^2z+5z^2) is -20xy^2z+4y

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(\partial)/(\partial x)(4xy-10x^2y^2z+5z^2)

\frac{\partial}{\partial x} (4 x y - 10 x^{2} y^{2} z + 5 z^{2})

- 20 x y^{2} z + 4 y

Solution steps

\frac{\partial}{\partial x} (4 x y - 10 x^{2} y^{2} z + 5 z^{2})

Treat

y, z

as constants

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial x} (4 x y) - \frac{\partial}{\partial x} (10 x^{2} y^{2} z) + \frac{\partial}{\partial x} (5 z^{2})

\frac{\partial}{\partial x} (4 x y) = 4 y

\frac{\partial}{\partial x} (10 x^{2} y^{2} z) = 20 y^{2} z x

\frac{\partial}{\partial x} (5 z^{2}) = 0

= 4 y - 20 y^{2} z x + 0

Simplify

= - 20 x y^{2} z + 4 y

y^{\frac{4}{5}}

x \to 1 - lim (\frac{2 x}{x ^{3} - 1})

\frac{d}{d x} ((x^{4} + 8)^{\frac{1}{2}})

\frac{\partial}{\partial x} (y^{\frac{1}{3}})

\frac{d}{d x} ((x^{2} - 1)^{2})

What is (\partial)/(\partial x)(4xy-10x^2y^2z+5z^2) ?
The answer to (\partial)/(\partial x)(4xy-10x^2y^2z+5z^2) is -20xy^2z+4y