What is (\partial)/(\partial x)(6x(x^2+y^2+z^2)^2) ?

The answer to (\partial)/(\partial x)(6x(x^2+y^2+z^2)^2) is 6(5x^4+6x^2y^2+6x^2z^2+y^4+z^4+2y^2z^2)

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(\partial)/(\partial x)(6x(x^2+y^2+z^2)^2)

\frac{\partial}{\partial x} (6 x (x^{2} + y^{2} + z^{2})^{2})

6 (5 x^{4} + 6 x^{2} y^{2} + 6 x^{2} z^{2} + y^{4} + z^{4} + 2 y^{2} z^{2})

Solution steps

\frac{\partial}{\partial x} (6 x (x^{2} + y^{2} + z^{2})^{2})

Treat

y, z

as constants

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 6 \frac{\partial}{\partial x} (x (x^{2} + y^{2} + z^{2})^{2})

Apply the Product Rule:

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

f = x, g = (x^{2} + y^{2} + z^{2})^{2}

= 6 (\frac{\partial}{\partial x} (x) (x^{2} + y^{2} + z^{2})^{2} + \frac{\partial}{\partial x} ((x^{2} + y^{2} + z^{2})^{2}) x)

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

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

= 6 (1 \cdot (x^{2} + y^{2} + z^{2})^{2} + 4 x (x^{2} + y^{2} + z^{2}) x)

Simplify

6 (1 \cdot (x^{2} + y^{2} + z^{2})^{2} + 4 x (x^{2} + y^{2} + z^{2}) x) : 6 (5 x^{4} + 6 x^{2} y^{2} + 6 x^{2} z^{2} + y^{4} + z^{4} + 2 y^{2} z^{2})

= 6 (5 x^{4} + 6 x^{2} y^{2} + 6 x^{2} z^{2} + y^{4} + z^{4} + 2 y^{2} z^{2})

\int \frac{( 2 x ^{2} + 2 )}{( x ^{2} - 2 x + 2 ) ^{2}} d x

f (x) = 3 x^{4} - x^{3} - 5 x^{2} + 7, at x = 2

\int x^{4} (7 - \frac{x ^{5}}{10})^{3} d x

\int - x sin (y) d y

\int 3 y d y

What is (\partial)/(\partial x)(6x(x^2+y^2+z^2)^2) ?
The answer to (\partial)/(\partial x)(6x(x^2+y^2+z^2)^2) is 6(5x^4+6x^2y^2+6x^2z^2+y^4+z^4+2y^2z^2)