Differentiate x^2 ln(x): animated lesson

1. 1. The problem
   Finding a derivative

   $\frac{d}{dx}\left[{x}^{2} \ln\left( x\right)\right]$
2. 2. Step by step
   1. $$\frac{d}{dx}\left[{x}^{2} \ln\left( x\right)\right] = x \left(2\ln\left( x\right)+1\right)$$
      Product rule: (uv)′ = u′v + uv′
   2. $$\frac{d}{dx}\left[\ln\left( x\right)\right] = \frac{1}{x}$$
      d/dx[ln x] = 1/x
   3. $$\frac{d}{dx}\left[{x}^{2}\right] = 2x$$
      Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
   4. $$f'(x) = x \left(2\ln\left( x\right)+1\right)$$
      Combine and simplify
3. 3. The answer
   $$f'(x) = x \left(2\ln\left( x\right)+1\right)$$

   f'(x) = x(2ln(x) + 1)

4. 4. Check
   • ✓Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Differentiate with the power, product, quotient and chain rules, including higher derivatives.

• Video lesson

  $\frac{d}{dx}\left[{x}^{-2}\right]$

  ▶0:43
  CalculusCollege2 steps

  Find d/dx x⁻²

  Derivatives

• Video lesson

  $\frac{d}{dx}\left[\frac{x}{x+1}\right]$

  ▶0:45
  CalculusCollege3 steps

  Find the derivative of x/(x+1)

  Derivatives

• Video lesson

  $\frac{d}{dx}\left[{2}^{x}\right]$

  ▶0:42
  CalculusCollege2 steps

  Find d/dx(2ˣ

  Derivatives

• Video lesson

  $\frac{d}{dx}\left[\tan^{-1}\left({x}^{2}+ x+1\right) \ln\left({e}^{x}+1\right)\right]$

  ▶1:25
  CalculusCollege9 steps

  Derivative of (arctan(x²+x+1))(ln(eˣ+1))

  Derivatives

• Video lesson

  $\frac{d}{dx}\left[{\left(2x\right)}^{3}+\ln\left({x}^{2}+1\right)\right]$

  ▶1:07
  CalculusCollege7 steps

  Derivative of ((2x)³+(ln(x²+1))

  Derivatives

• Video lesson

  $\frac{d}{dx}\left[{e}^{{e}^{x}+1} \cos\left({x}^{2}+1\right)\right]$

  ▶1:22
  CalculusCollege8 steps

  Derivative of (e^(eˣ+1))(cos(x²+1))

  Derivatives

• All derivatives lessons
• Browse the whole lesson library
• Make a lesson from your own problem
• Step-by-step solver: check your own answer.