What is (\partial)/(\partial x)((x^3)sqrt(x^2+9)) ?

The answer to (\partial)/(\partial x)((x^3)sqrt(x^2+9)) is (4x^4+27x^2)/(sqrt(x^2+9))

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(\partial)/(\partial x)((x^3)sqrt(x^2+9))

\frac{\partial}{\partial x} ((x^{3}) x^{2} + 9)

\frac{4 x ^{4} + 27 x ^{2}}{x ^{2} + 9}

Solution steps

\frac{\partial}{\partial x} ((x^{3}) x^{2} + 9)

Apply the Product Rule:

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

f = x^{3}, g = x^{2} + 9

= \frac{\partial}{\partial x} (x^{3}) x^{2} + 9 + \frac{\partial}{\partial x} (x^{2} + 9) x^{3}

\frac{\partial}{\partial x} (x^{3}) = 3 x^{2}

\frac{\partial}{\partial x} (x^{2} + 9) = \frac{x}{x ^{2} + 9}

= 3 x^{2} x^{2} + 9 + \frac{x}{x ^{2} + 9} x^{3}

Simplify

3 x^{2} x^{2} + 9 + \frac{x}{x ^{2} + 9} x^{3} : \frac{4 x ^{4} + 27 x ^{2}}{x ^{2} + 9}

= \frac{4 x ^{4} + 27 x ^{2}}{x ^{2} + 9}

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

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

\frac{d y}{d x} = 2 y + 1

n = 0 \sum \infty \frac{2}{1.1 5 ^{n}}

y^{^{''}} + 25 y = - 30 sec (5 t)

What is (\partial)/(\partial x)((x^3)sqrt(x^2+9)) ?
The answer to (\partial)/(\partial x)((x^3)sqrt(x^2+9)) is (4x^4+27x^2)/(sqrt(x^2+9))