derivative of y=3x+4/x — step-by-step solution

We save the definition:

y = 3 x + \frac{4}{x}

To ease differentiation, the expression

\frac{4}{x}

is turned into

4 x^{-1}

Let's differentiate:

\frac{d}{d x} (3 x + 4 x^{-1})

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

\frac{d}{d x} (3 x + 4 x^{-1})

\frac{d}{d x} (3 x) + \frac{d}{d x} (4 x^{-1})

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

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

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

3

\frac{d}{d x} (4 x^{-1})

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

■The derivative of x^n is n*x^(n-1):

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

-1 \cdot x^{-2}

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

ⓘ Multiplications are done before additions and subtractions. A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

4 \cdot (-1)

-4

3 - 4 x^{-2}

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

4 x^{-2}

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

▶Derivative:

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