What is the derivative of ax*ln(ax^2+3) ?

The derivative of ax*ln(ax^2+3) is a(ln(ax^2+3)+(2ax^2)/(ax^2+3))

What is the first derivative of ax*ln(ax^2+3) ?

The first derivative of ax*ln(ax^2+3) is a(ln(ax^2+3)+(2ax^2)/(ax^2+3))

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\frac{d}{d x} (a x \cdot ln (a x^{2} + 3))

a (ln (a x^{2} + 3) + \frac{2 a x ^{2}}{a x ^{2} + 3})

Solution steps

\frac{d}{d x} (a x ln (a x^{2} + 3))

Take the constant out:

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

= a \frac{d}{d x} (x ln (a x^{2} + 3))

Apply the Product Rule:

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

f = x, g = ln (a x^{2} + 3)

= a (\frac{d x}{d x} ln (a x^{2} + 3) + \frac{d}{d x} (ln (a x^{2} + 3)) x)

\frac{d x}{d x} = 1

\frac{d}{d x} (ln (a x^{2} + 3)) = \frac{2 a x}{a x ^{2} + 3}

= a (1 \cdot ln (a x^{2} + 3) + \frac{2 a x}{a x ^{2} + 3} x)

Simplify

a (1 \cdot ln (a x^{2} + 3) + \frac{2 a x}{a x ^{2} + 3} x) : a (ln (a x^{2} + 3) + \frac{2 a x ^{2}}{a x ^{2} + 3})

= a (ln (a x^{2} + 3) + \frac{2 a x ^{2}}{a x ^{2} + 3})

\int \frac{cos ( x )}{sin ^{3} ( x )} d x

\int \frac{x + 1}{x} d x

\frac{\partial}{\partial x} (\frac{x - y}{x})

\frac{d}{d x} (ln (\frac{x}{4}) (x - 4))

\int \frac{sin ( x )}{cot ( x )} d x

What is the derivative of ax*ln(ax^2+3) ?
The derivative of ax*ln(ax^2+3) is a(ln(ax^2+3)+(2ax^2)/(ax^2+3))
What is the first derivative of ax*ln(ax^2+3) ?
The first derivative of ax*ln(ax^2+3) is a(ln(ax^2+3)+(2ax^2)/(ax^2+3))