derivative of 5e^x+2/(\sqrt[3]{x})

To simplify differentiation, we rewrite

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

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

Let's perform the differentiation:

\frac{d}{d x} (5 e^{x} + 2 x^{- \frac{1}{3}})

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

\frac{d}{d x} (5 e^{x} + 2 x^{- \frac{1}{3}})

\frac{d}{d x} (5 e^{x}) + \frac{d}{d x} (2 x^{- \frac{1}{3}})

\frac{d}{d x} (5 e^{x}) + \frac{d}{d x} (2 x^{- \frac{1}{3}})

Apply the differentiation rule:

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

with f =

x

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

\frac{d}{d x} (5 e^{x})

5 \cdot \frac{d}{d x} (e^{x})

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

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

■The derivative of exp(x) is exp(x):

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

e^{x}

■The derivative of x^n is n*x^(n-1):

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

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

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

ⓘ Multiplications are done before additions and subtractions:

2 \cdot (- \frac{1}{3})

- \frac{2}{3}

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

ⓘ Multiplying a fraction by a term multiplies its numerator:

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

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

- \frac{2}{3 \sqrt[3]{x^{4}}}

▶Derivative:

5 e^{x} - \frac{2}{3 \sqrt[3]{x^{4}}}