WordGPT

Derivative of x51xx^{5} \frac{1}{x}

The derivative of x51xx^{5} \frac{1}{x} is 5x41x+x5āˆ’1x25 x^{4} \frac{1}{x} + x^{5} \frac{-1}{x^{2}}.

f(x) =
ddx[x51x]=5x41x+x5āˆ’1x2\frac{d}{dx}\left[x^{5} \frac{1}{x}\right] = 5 x^{4} \frac{1}{x} + x^{5} \frac{-1}{x^{2}}

Type any function of x; the derivative is computed symbolically in your browser.

Explain it step by step in chatMake a short video of it
Graph of f(x) = x^5 * 1 / x and its derivative-3-2-1123-60-40-2020406080slope 4f(x)f'(x)tangent at x = 1
Graph of f(x) = x^5 * 1 / x and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 4.

How to work it out

  1. 1

    We differentiate f(x)=x51xf(x) = x^{5} \frac{1}{x} with respect to x. The rule that applies is the product rule.

    f(x)=x51xf(x) = x^{5} \frac{1}{x}
  2. 2

    Product rule: (uv)ā€²=uā€²v+uvā€²(uv)' = u'v + uv'. Let u=x5u = x^{5} and v=1xv = \frac{1}{x}.

    u=x5,v=1xu = x^{5},\quad v = \frac{1}{x}
  3. 3

    Differentiate each piece:

    uā€²=5x4,vā€²=āˆ’1x2u' = 5 x^{4},\quad v' = \frac{-1}{x^{2}}
  4. 4

    Put them in the rule:

    fā€²(x)=(5x4)(1x)+(x5)(āˆ’1x2)f'(x) = \left(5 x^{4}\right)\left(\frac{1}{x}\right) + \left(x^{5}\right)\left(\frac{-1}{x^{2}}\right)
  5. 5

    Simplify:

    fā€²(x)=5x41x+x5āˆ’1x2f'(x) = 5 x^{4} \frac{1}{x} + x^{5} \frac{-1}{x^{2}}
  6. 6

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

    fā€²ā€²(x)=20x31x+5x4āˆ’1x2+5x4āˆ’1x2+x5āˆ’((āˆ’2)x)(x2)2f''(x) = 20 x^{3} \frac{1}{x} + 5 x^{4} \frac{-1}{x^{2}} + 5 x^{4} \frac{-1}{x^{2}} + x^{5} \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[x^{5} \frac{1}{x}\right] = 5 x^{4} \frac{1}{x} + x^{5} \frac{-1}{x^{2}} \]
\[ f(x) = x^{5} \frac{1}{x} \]
\[ u = x^{5},\quad v = \frac{1}{x} \]
\[ u' = 5 x^{4},\quad v' = \frac{-1}{x^{2}} \]
\[ f'(x) = \left(5 x^{4}\right)\left(\frac{1}{x}\right) + \left(x^{5}\right)\left(\frac{-1}{x^{2}}\right) \]
\[ f'(x) = 5 x^{4} \frac{1}{x} + x^{5} \frac{-1}{x^{2}} \]
\[ f''(x) = 20 x^{3} \frac{1}{x} + 5 x^{4} \frac{-1}{x^{2}} + 5 x^{4} \frac{-1}{x^{2}} + x^{5} \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}} \]

Values of f and f' for x5x^{5} * 1 / x

xf(x)f'(x)
-216-32
-11-4
0.50.06250.5
114
21632
381108

Questions

What is the derivative of x5x^{5} * 1 / x?

The derivative of x51xx^{5} \frac{1}{x} is 5x41x+x5āˆ’1x25 x^{4} \frac{1}{x} + x^{5} \frac{-1}{x^{2}}.

What is the second derivative of x5x^{5} * 1 / x?

20 * x3x^{3} * 1 / x + 5 * x4x^{4} * -1 / x2x^{2} + 5 * x4x^{4} * -1 / x2x^{2} + x5x^{5} * -(-2x) / (x2x^{2})^2.

Which rule do you use for x5x^{5} * 1 / x?

The product rule.

What is the slope of x5x^{5} * 1 / x at x = 1?

4.

Related derivatives

    Derivative of x^5 * 1 / x: 5 * x^4 * 1 / x + x^5 * -1 / x^2 (with steps) | WordGPT