derivative of ln(3x+x^2) — step-by-step solution

Let's differentiate:

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

We differentiate using:

\frac{d}{d x} (\ln (f)) = \frac{d}{d x} (f) \cdot \frac{1}{f}

(where f =

3 x + x^{2}

)

ⓘ The derivative of ln(x) is 1/x:

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

\frac{d}{d x} (3 x + x^{2}) \cdot \frac{1}{3 x + x^{2}}

\frac{d}{d x} (3 x + x^{2}) \cdot \frac{1}{3 x + x^{2}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} (3 x + x^{2}) \cdot \frac{1}{3 x + x^{2}}

\frac{\frac{d}{d x} (3 x + x^{2}) \cdot 1}{3 x + x^{2}}

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

■What the terms have in common is factored out, so that the fraction can be simplified:

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

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

■The derivative of a sum is the sum of the derivatives:

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

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

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

ⓘ A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (3 x)

3

■The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{2})

2 x

\frac{3 + 2 x}{x \cdot (3 + x)}

▶Derivative:

\frac{2 x + 3}{x \cdot (x + 3)}