What is (\partial)/(\partial x)(x(x^2+y^2+z^2-1)) ?

The answer to (\partial)/(\partial x)(x(x^2+y^2+z^2-1)) is 3x^2+y^2+z^2-1

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(\partial)/(\partial x)(x(x^2+y^2+z^2-1))

\frac{\partial}{\partial x} (x (x^{2} + y^{2} + z^{2} - 1))

3 x^{2} + y^{2} + z^{2} - 1

Solution steps

\frac{\partial}{\partial x} (x (x^{2} + y^{2} + z^{2} - 1))

Treat

y, z

as constants

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = x^{2} + y^{2} + z^{2} - 1

= \frac{\partial}{\partial x} (x) (x^{2} + y^{2} + z^{2} - 1) + \frac{\partial}{\partial x} (x^{2} + y^{2} + z^{2} - 1) x

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (x^{2} + y^{2} + z^{2} - 1) = 2 x

= 1 \cdot (x^{2} + y^{2} + z^{2} - 1) + 2 xx

Simplify

1 \cdot (x^{2} + y^{2} + z^{2} - 1) + 2 xx : 3 x^{2} + y^{2} + z^{2} - 1

= 3 x^{2} + y^{2} + z^{2} - 1

x \to 1 - lim (x^{2} - 5)

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

\frac{d}{d x} y - y = 2

\int \frac{2 x + 1}{x ^{2} + x - 2} d x

\frac{d}{d x} (\frac{x}{y ^{2} + x ^{2}})

What is (\partial)/(\partial x)(x(x^2+y^2+z^2-1)) ?
The answer to (\partial)/(\partial x)(x(x^2+y^2+z^2-1)) is 3x^2+y^2+z^2-1