derivative of x^4ln(x) — step-by-step solution

1) Let's perform the differentiation:

\frac{d}{d x} (x^{4} \cdot \ln (x))

We differentiate

u1

x^{4}

and

v2

\ln (x)

using:

\frac{d}{d x} (u1 \cdot v2) = \frac{d}{d x} (u1) \cdot v2 + u1 \cdot \frac{d}{d x} (v2)

2) Let's differentiate

u1

\frac{d}{d x} (x^{4})

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{4})

4 x^{3}

4 x^{3}

3) Let's differentiate

v2

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

ⓘ The derivative of ln(x) is 1/x:

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

\frac{1}{x}

\frac{1}{x}

4) Let's evaluate the resulting derivative (

u1 \cdot v2

)':

4 x^{3} \cdot \ln (x) + x^{4} \cdot \frac{1}{x}

ⓘ Multiplying a fraction by a term multiplies its numerator:

x^{4} \cdot \frac{1}{x}

\frac{x^{4} \cdot 1}{x}

\frac{x^{4}}{x}

■The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor:

\frac{x^{4}}{x}

x^{3}

4 x^{3} \cdot \ln (x) + x^{3}

5) You can then resume the differentiation:

ⅰ

\frac{d}{d x} (x^{4} \cdot \ln (x))

4 x^{3} \cdot \ln (x) + x^{3}

4 x^{3} \cdot \ln (x) + x^{3}

x^{3} \cdot (4 \cdot \ln (x) + 1)

▶Derivative:

x^{3} \cdot (4 \cdot \ln (x) + 1)