What is (\partial)/(\partial x)(3x^2(5x+7y)) ?

The answer to (\partial)/(\partial x)(3x^2(5x+7y)) is 3(15x^2+14xy)

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(\partial)/(\partial x)(3x^2(5x+7y))

\frac{\partial}{\partial x} (3 x^{2} (5 x + 7 y))

3 (15 x^{2} + 14 x y)

Solution steps

\frac{\partial}{\partial x} (3 x^{2} (5 x + 7 y))

Treat

y

as a constant

Take the constant out:

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

= 3 \frac{\partial}{\partial x} (x^{2} (5 x + 7 y))

Apply the Product Rule:

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

f = x^{2}, g = 5 x + 7 y

= 3 (\frac{\partial}{\partial x} (x^{2}) (5 x + 7 y) + \frac{\partial}{\partial x} (5 x + 7 y) x^{2})

\frac{\partial}{\partial x} (x^{2}) = 2 x

\frac{\partial}{\partial x} (5 x + 7 y) = 5

= 3 (2 x (5 x + 7 y) + 5 x^{2})

Expand

2 x (5 x + 7 y) + 5 x^{2} : 15 x^{2} + 14 x y

= 3 (15 x^{2} + 14 x y)

What is (\partial)/(\partial x)(3x^2(5x+7y)) ?
The answer to (\partial)/(\partial x)(3x^2(5x+7y)) is 3(15x^2+14xy)