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

To ease differentiation, the expression

\frac{2}{2 x}

is turned into

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

Let's differentiate:

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

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{2}{2}

1

■Multiplying by 1 changes nothing:

1 \cdot x^{-1}

x^{-1}

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

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

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

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

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

Apply the differentiation rule:

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

with f =

x

ⓘ The derivative of x^n is n*x^(n-1):

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

-1 \cdot x^{-2}

2 x + 1 - 1 \cdot x^{-2}

ⓘ Multiplying by 1 changes nothing:

1 \cdot x^{-2}

x^{-2}

2 x + 1 - x^{-2}

ⓘ A negative exponent moves the power to the other side of the fraction bar, for a nonzero base:

x^{-2}

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

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

▶Derivative:

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