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Derivative of x2xx^{2} \sqrt{x}

The derivative of x2xx^{2} \sqrt{x} is 2xx+x212x2 x \sqrt{x} + x^{2} \frac{1}{2 \sqrt{x}}.

f(x) =
ddx[x2x]=2xx+x212x\frac{d}{dx}\left[x^{2} \sqrt{x}\right] = 2 x \sqrt{x} + x^{2} \frac{1}{2 \sqrt{x}}

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Graph of f(x) = x^2 * sqrt(x) and its derivative-3-2-1123-10-551015slope 2.5f(x)f'(x)tangent at x = 1
Graph of f(x) = x^2 * sqrt(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 2.5.

How to work it out

  1. 1

    We differentiate f(x)=x2xf(x) = x^{2} \sqrt{x} with respect to x. The rule that applies is the product rule.

    f(x)=x2xf(x) = x^{2} \sqrt{x}
  2. 2

    Product rule: (uv)ā€²=uā€²v+uvā€²(uv)' = u'v + uv'. Let u=x2u = x^{2} and v=xv = \sqrt{x}.

    u=x2,v=xu = x^{2},\quad v = \sqrt{x}
  3. 3

    Differentiate each piece:

    uā€²=2x,vā€²=12xu' = 2 x,\quad v' = \frac{1}{2 \sqrt{x}}
  4. 4

    Put them in the rule:

    fā€²(x)=(2x)(x)+(x2)(12x)f'(x) = \left(2 x\right)\left(\sqrt{x}\right) + \left(x^{2}\right)\left(\frac{1}{2 \sqrt{x}}\right)
  5. 5

    Simplify:

    fā€²(x)=2xx+x212xf'(x) = 2 x \sqrt{x} + x^{2} \frac{1}{2 \sqrt{x}}
  6. 6

    Differentiating once more gives the second derivative, which tells you about curvature:

    fā€²ā€²(x)=2x+2x12x+2x12x+x2āˆ’(212x)(2x)2f''(x) = 2 \sqrt{x} + 2 x \frac{1}{2 \sqrt{x}} + 2 x \frac{1}{2 \sqrt{x}} + x^{2} \frac{-\left(2 \frac{1}{2 \sqrt{x}}\right)}{\left(2 \sqrt{x}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[x^{2} \sqrt{x}\right] = 2 x \sqrt{x} + x^{2} \frac{1}{2 \sqrt{x}} \]
\[ f(x) = x^{2} \sqrt{x} \]
\[ u = x^{2},\quad v = \sqrt{x} \]
\[ u' = 2 x,\quad v' = \frac{1}{2 \sqrt{x}} \]
\[ f'(x) = \left(2 x\right)\left(\sqrt{x}\right) + \left(x^{2}\right)\left(\frac{1}{2 \sqrt{x}}\right) \]
\[ f'(x) = 2 x \sqrt{x} + x^{2} \frac{1}{2 \sqrt{x}} \]
\[ f''(x) = 2 \sqrt{x} + 2 x \frac{1}{2 \sqrt{x}} + 2 x \frac{1}{2 \sqrt{x}} + x^{2} \frac{-\left(2 \frac{1}{2 \sqrt{x}}\right)}{\left(2 \sqrt{x}\right)^{2}} \]

Values of f and f' for x2x^{2} * sqrt(x)

xf(x)f'(x)
0.50.17677670.88388348
112.5
25.65685427.0710678
315.58845712.990381

Questions

What is the derivative of x2x^{2} * sqrt(x)?

The derivative of x2xx^{2} \sqrt{x} is 2xx+x212x2 x \sqrt{x} + x^{2} \frac{1}{2 \sqrt{x}}.

What is the second derivative of x2x^{2} * sqrt(x)?

2sqrt(x) + 2x * 1 / (2sqrt(x)) + 2x * 1 / (2sqrt(x)) + x2x^{2} * -(2 * 1 / (2sqrt(x))) / (2sqrt(x))^2.

Which rule do you use for x2x^{2} * sqrt(x)?

The product rule.

What is the slope of x2x^{2} * sqrt(x) at x = 1?

2.5.

Related derivatives

    Derivative of x^2 * sqrt(x): 2x * sqrt(x) + x^2 * 1 / (2sqrt(x)) (with steps) | WordGPT