derivative of 3x^4-5x+2\sqrt[3]{x}

Let's differentiate:

\frac{d}{d x} (3 x^{4} - 5 x + 2 \sqrt[3]{x})

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

\frac{d}{d x} (3 x^{4} - 5 x + 2 \sqrt[3]{x})

\frac{d}{d x} (3 x^{4}) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (2 \sqrt[3]{x})

\frac{d}{d x} (3 x^{4}) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (2 \sqrt[3]{x})

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

\frac{d}{d x} (3 x^{4})

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

\frac{d}{d x} (- 5 x)

-5

\frac{d}{d x} (2 \sqrt[3]{x})

2 \cdot \frac{d}{d x} (\sqrt[3]{x})

■The derivative of x^n is n*x^(n-1):

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

4 x^{3}

\frac{d}{d x} (\sqrt[3]{x})

\frac{1}{3} \cdot x^{- \frac{2}{3}}

3 \cdot 4 x^{3} - 5 + 2 \cdot \frac{1}{3} \cdot x^{- \frac{2}{3}}

ⓘ Multiplications are done before additions and subtractions:

3 \cdot 4

12

2 \cdot \frac{1}{3}

\frac{2}{3}

12 x^{3} - 5 + \frac{2}{3} \cdot x^{- \frac{2}{3}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{2}{3} \cdot x^{- \frac{2}{3}}

\frac{2 x^{- \frac{2}{3}}}{3}

■The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

\frac{2 x^{- \frac{2}{3}}}{3}

\frac{2}{3 \sqrt[3]{x^{2}}}

12 x^{3} - 5 + \frac{2}{3 \sqrt[3]{x^{2}}}

▶Derivative:

12 x^{3} + \frac{2}{3 \sqrt[3]{x^{2}}} - 5