What is the d/(dt)((x)(3x+2y+4z)) ?

The d/(dt)((x)(3x+2y+4z)) is 6x(dx)/(dt)+2y(dx)/(dt)+4z(dx)/(dt)+2x(dy)/(dt)+4x(dz)/(dt)

What is the first d/(dt)((x)(3x+2y+4z)) ?

The first d/(dt)((x)(3x+2y+4z)) is 6x(dx)/(dt)+2y(dx)/(dt)+4z(dx)/(dt)+2x(dy)/(dt)+4x(dz)/(dt)

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\frac{d}{d t} ((x) (3 x + 2 y + 4 z))

6 x \frac{d x}{d t} + 2 y \frac{d x}{d t} + 4 z \frac{d x}{d t} + 2 x \frac{d y}{d t} + 4 x \frac{d z}{d t}

Solution steps

\frac{d}{d t} ((x) (3 x + 2 y + 4 z))

Apply the Product Rule:

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

f = x, g = 3 x + 2 y + 4 z

= \frac{d x}{d t} (3 x + 2 y + 4 z) + \frac{d}{d t} (3 x + 2 y + 4 z) x

\frac{d}{d t} (3 x + 2 y + 4 z) = 3 \frac{d x}{d t} + 2 \frac{d y}{d t} + 4 \frac{d z}{d t}

= \frac{d x}{d t} (3 x + 2 y + 4 z) + (3 \frac{d x}{d t} + 2 \frac{d y}{d t} + 4 \frac{d z}{d t}) x

Simplify

\frac{d x}{d t} (3 x + 2 y + 4 z) + (3 \frac{d x}{d t} + 2 \frac{d y}{d t} + 4 \frac{d z}{d t}) x : 6 x \frac{d x}{d t} + 2 y \frac{d x}{d t} + 4 z \frac{d x}{d t} + 2 x \frac{d y}{d t} + 4 x \frac{d z}{d t}

= 6 x \frac{d x}{d t} + 2 y \frac{d x}{d t} + 4 z \frac{d x}{d t} + 2 x \frac{d y}{d t} + 4 x \frac{d z}{d t}

4 (9 + x^{2}) + 5 (4 - x)

(1 + sec (θ))^{2}

\frac{( x + 5 ) ^{3}}{3}

\frac{e ^{2 x} ( 5 e ^{3 x} - 2 )}{25}

(\frac{1}{2} - 3 x) (\frac{1}{2} + 3 x)

What is the d/(dt)((x)(3x+2y+4z)) ?
The d/(dt)((x)(3x+2y+4z)) is 6x(dx)/(dt)+2y(dx)/(dt)+4z(dx)/(dt)+2x(dy)/(dt)+4x(dz)/(dt)
What is the first d/(dt)((x)(3x+2y+4z)) ?
The first d/(dt)((x)(3x+2y+4z)) is 6x(dx)/(dt)+2y(dx)/(dt)+4z(dx)/(dt)+2x(dy)/(dt)+4x(dz)/(dt)