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

The answer to (\partial)/(\partial x)(5xln(x^2+y)) is 5(ln(x^2+y)+(2x^2)/(x^2+y))

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(\partial)/(\partial x)(5xln(x^2+y))

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

5 (ln (x^{2} + y) + \frac{2 x ^{2}}{x ^{2} + y})

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

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

= 5 (1 \cdot ln (x^{2} + y) + \frac{2 x}{x ^{2} + y} x)

Simplify

5 (1 \cdot ln (x^{2} + y) + \frac{2 x}{x ^{2} + y} x) : 5 (ln (x^{2} + y) + \frac{2 x ^{2}}{x ^{2} + y})

= 5 (ln (x^{2} + y) + \frac{2 x ^{2}}{x ^{2} + y})

x \to 0 lim (x^{2} - 6)

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What is (\partial)/(\partial x)(5xln(x^2+y)) ?
The answer to (\partial)/(\partial x)(5xln(x^2+y)) is 5(ln(x^2+y)+(2x^2)/(x^2+y))