derivative of x^2sqrt(x)-5x — step-by-step solution

Let's differentiate:

\frac{d}{d x} (x^{2} \sqrt{x} - 5 x)

ⓘ Multiplying powers with the same base adds the exponents:

x^{2} \sqrt{x}

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

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

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

2 + \frac{1}{2}

\frac{2 \cdot 2 + 1}{2}

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

ⓘ Multiplications are done before additions and subtractions:

2 \cdot 2

4

\frac{d}{d x} (x^{\frac{4 + 1}{2}} - 5 x)

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

4 + 1

5

\frac{d}{d x} (x^{\frac{5}{2}} - 5 x)

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

\frac{d}{d x} (x^{\frac{5}{2}} - 5 x)

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

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

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{\frac{5}{2}})

\frac{5}{2} \cdot x^{\frac{3}{2}}

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

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

-5

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

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{5}{2} \cdot x^{\frac{3}{2}}

\frac{5 x^{\frac{3}{2}}}{2}

\frac{5 x^{\frac{3}{2}}}{2} - 5

We factor out

5

, the greatest common divisor of the coefficients:

\frac{5 x^{\frac{3}{2}}}{2} - 5

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

Factor identified:

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

▶Derivative:

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