What is the derivative of (4x^2+9(8x+x^{1/2})) ?

The derivative of (4x^2+9(8x+x^{1/2})) is (20x^2+192x^{5/2}+9)/(2x^{1/2)}+72

What is the first derivative of (4x^2+9(8x+x^{1/2})) ?

The first derivative of (4x^2+9(8x+x^{1/2})) is (20x^2+192x^{5/2}+9)/(2x^{1/2)}+72

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\frac{d}{d x} ((4 x^{2} + 9) (8 x + x^{\frac{1}{2}}))

\frac{20 x ^{2} + 192 x ^{\frac{5}{2}} + 9}{2 x ^{\frac{1}{2}}} + 72

Solution steps

\frac{d}{d x} ((4 x^{2} + 9) (8 x + x^{\frac{1}{2}}))

Apply the Product Rule:

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

f = 4 x^{2} + 9, g = 8 x + x^{\frac{1}{2}}

= \frac{d}{d x} (4 x^{2} + 9) (8 x + x^{\frac{1}{2}}) + \frac{d}{d x} (8 x + x^{\frac{1}{2}}) (4 x^{2} + 9)

\frac{d}{d x} (4 x^{2} + 9) = 8 x

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

= 8 x (8 x + x^{\frac{1}{2}}) + (8 + \frac{1}{2 x ^{\frac{1}{2}}}) (4 x^{2} + 9)

Simplify

8 x (8 x + x^{\frac{1}{2}}) + (8 + \frac{1}{2 x ^{\frac{1}{2}}}) (4 x^{2} + 9) : \frac{20 x ^{2} + 192 x ^{\frac{5}{2}} + 9}{2 x ^{\frac{1}{2}}} + 72

= \frac{20 x ^{2} + 192 x ^{\frac{5}{2}} + 9}{2 x ^{\frac{1}{2}}} + 72

y = 7 x^{\frac{1}{2}} + x^{\frac{3}{2}}

\int \frac{5 x}{3 x ^{2} + 4} d x

n = 0 \sum \infty 5 (\frac{3}{2})^{n}

\int x^{2} 8 x + 9 d x

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

What is the derivative of (4x^2+9(8x+x^{1/2})) ?
The derivative of (4x^2+9(8x+x^{1/2})) is (20x^2+192x^{5/2}+9)/(2x^{1/2)}+72
What is the first derivative of (4x^2+9(8x+x^{1/2})) ?
The first derivative of (4x^2+9(8x+x^{1/2})) is (20x^2+192x^{5/2}+9)/(2x^{1/2)}+72