WordGPT

Derivative of xsin⁡(x)\sqrt{x} \sin\left(x\right)

The derivative of xsin⁡(x)\sqrt{x} \sin\left(x\right) is sin⁡(x)2x+xcos⁡(x)\frac{\sin\left(x\right)}{2 \sqrt{x}} + \sqrt{x} \cos\left(x\right).

f(x) =
ddx[xsin⁡(x)]=sin⁡(x)2x+xcos⁡(x)\frac{d}{dx}\left[\sqrt{x} \sin\left(x\right)\right] = \frac{\sin\left(x\right)}{2 \sqrt{x}} + \sqrt{x} \cos\left(x\right)

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) = sqrt(x) * sin(x) and its derivative-3-2-1123-3-2-1123slope 0.961f(x)f'(x)tangent at x = 1
Graph of f(x) = sqrt(x) * sin(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 0.96104.

How to work it out

  1. 1

    We differentiate f(x)=xsin⁡(x)f(x) = \sqrt{x} \sin\left(x\right) with respect to x. The rule that applies is the product rule.

    f(x)=xsin⁡(x)f(x) = \sqrt{x} \sin\left(x\right)
  2. 2

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. Let u=xu = \sqrt{x} and v=sin⁡(x)v = \sin\left(x\right).

    u=x,v=sin⁡(x)u = \sqrt{x},\quad v = \sin\left(x\right)
  3. 3

    Differentiate each piece:

    u′=12x,v′=cos⁡(x)u' = \frac{1}{2 \sqrt{x}},\quad v' = \cos\left(x\right)
  4. 4

    Put them in the rule:

    f′(x)=(12x)(sin⁡(x))+(x)(cos⁡(x))f'(x) = \left(\frac{1}{2 \sqrt{x}}\right)\left(\sin\left(x\right)\right) + \left(\sqrt{x}\right)\left(\cos\left(x\right)\right)
  5. 5

    Simplify:

    f′(x)=sin⁡(x)2x+xcos⁡(x)f'(x) = \frac{\sin\left(x\right)}{2 \sqrt{x}} + \sqrt{x} \cos\left(x\right)
  6. 6

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

    f′′(x)=cos⁡(x)2x−sin⁡(x)212x(2x)2+cos⁡(x)2x+x−sin⁡(x)f''(x) = \frac{\cos\left(x\right) 2 \sqrt{x} - \sin\left(x\right) 2 \frac{1}{2 \sqrt{x}}}{\left(2 \sqrt{x}\right)^{2}} + \frac{\cos\left(x\right)}{2 \sqrt{x}} + \sqrt{x} -\sin\left(x\right)
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\sqrt{x} \sin\left(x\right)\right] = \frac{\sin\left(x\right)}{2 \sqrt{x}} + \sqrt{x} \cos\left(x\right) \]
\[ f(x) = \sqrt{x} \sin\left(x\right) \]
\[ u = \sqrt{x},\quad v = \sin\left(x\right) \]
\[ u' = \frac{1}{2 \sqrt{x}},\quad v' = \cos\left(x\right) \]
\[ f'(x) = \left(\frac{1}{2 \sqrt{x}}\right)\left(\sin\left(x\right)\right) + \left(\sqrt{x}\right)\left(\cos\left(x\right)\right) \]
\[ f'(x) = \frac{\sin\left(x\right)}{2 \sqrt{x}} + \sqrt{x} \cos\left(x\right) \]
\[ f''(x) = \frac{\cos\left(x\right) 2 \sqrt{x} - \sin\left(x\right) 2 \frac{1}{2 \sqrt{x}}}{\left(2 \sqrt{x}\right)^{2}} + \frac{\cos\left(x\right)}{2 \sqrt{x}} + \sqrt{x} -\sin\left(x\right) \]

Values of f and f' for sqrt(x) * sin(x)

xf(x)f'(x)
0.50.339005050.95954963
10.841470980.9610378
21.2859408-0.26703531
30.24442702-1.6739795

Questions

What is the derivative of sqrt(x) * sin(x)?

The derivative of xsin⁡(x)\sqrt{x} \sin\left(x\right) is sin⁡(x)2x+xcos⁡(x)\frac{\sin\left(x\right)}{2 \sqrt{x}} + \sqrt{x} \cos\left(x\right).

What is the second derivative of sqrt(x) * sin(x)?

(cos(x) * 2sqrt(x) - sin(x) * 2 * 1 / (2sqrt(x))) / (2sqrt(x))^2 + cos(x) / (2sqrt(x)) + sqrt(x) * -sin(x).

Which rule do you use for sqrt(x) * sin(x)?

The product rule.

What is the slope of sqrt(x) * sin(x) at x = 1?

0.9610378.

Related derivatives

    Derivative of sqrt(x) * sin(x): sin(x) / (2sqrt(x)) + sqrt(x) * cos(x) (with steps) | WordGPT