derivative of 3\sqrt[3]{x^2}-2x — step-by-step solution

Let's differentiate:

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

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

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

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

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

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

with f =

x^{2}

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

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

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

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

-2

■The derivative of x^n is n*x^(n-1):

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

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

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

ⓘ Multiplications are done before additions and subtractions:

3 \cdot \frac{1}{3}

1

■Multiplying by 1 changes nothing:

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

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

■A power of a power multiplies the exponents, (a^n)^m = a^(n*m):

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

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

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

ⓘ The derivative of x^n is n*x^(n-1):

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

2 x

■The minus sign is placed in front of the product:

2 \cdot (- \frac{2}{3})

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

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

ⓘ Multiplying a fraction by a number multiplies its numerator:

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

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

- \frac{4}{3}

■Multiplying powers with the same base adds the exponents:

x x^{- \frac{4}{3}}

x^{1 - \frac{4}{3}}

2 x^{1 - \frac{4}{3}} - 2

ⓘ To add fractions, put them over the same denominator, then add the numerators:

1 - \frac{4}{3}

\frac{3 - 4}{3}

2 x^{\frac{3 - 4}{3}} - 2

ⓘ The numerator and the denominator are computed before the division:

3 - 4

-1

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

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

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

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

▶Derivative:

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