Evaluate second derivative of cos(3x): animated lesson

1. 1. The problem
   Finding a derivative

   $\frac{d^{2}}{dx^{2}}\left[\cos\left(3x\right)\right]$
2. 2. Steps, part 1
   1. $$\frac{d}{dx}\left[\cos\left(3x\right)\right]$$
      Derivative number 1
   2. $$\frac{d}{dx}\left[\cos\left(3x\right)\right] = -\left(3\sin\left(3x\right)\right)$$
      Chain rule — d/dx[cos u] = −sin u · u′
   3. $$\frac{d}{dx}\left[3x\right] = 3$$
      Constant multiple rule
   4. $$f'(x) = -\left(3\sin\left(3x\right)\right)$$
      Simplify the first derivative
3. 3. Steps, part 2
   1. $$\frac{d}{dx}\left[-\left(3\sin\left(3x\right)\right)\right]$$
      Derivative number 2
   2. $$\frac{d}{dx}\left[3\sin\left(3x\right)\right] = 9\cos\left(3x\right)$$
      Constant multiple rule
   3. $$\frac{d}{dx}\left[\sin\left(3x\right)\right] = 3\cos\left(3x\right)$$
      Chain rule — d/dx[sin u] = cos u · u′
   4. $$\frac{d}{dx}\left[3x\right] = 3$$
      Constant multiple rule
4. 4. Steps, part 3
   1. $$f''(x) = -\left(9\cos\left(3x\right)\right)$$
      Simplify the second derivative
5. 5. The answer
   $$f''(x) = -\left(9\cos\left(3x\right)\right)$$

   f''(x) = -(9cos(3x))

6. 6. 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

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

  ▶0:48
  CalculusCollege3 steps

  Derivative of arctan(x²

  Derivatives

• Video lesson

  $\frac{d}{dx}\left[\sin^{-1}\left( x\right)\right]$

  ▶0:41
  CalculusCollege2 steps

  Differentiate arcsin(x)

  Derivatives

• Video lesson

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

  ▶0:40
  CalculusCollege2 steps

  Find the derivative of √(x)

  Derivatives

• Video lesson

  $\frac{d}{dx}\left[\log_{2}\left( x\right)\right]$

  ▶0:38
  CalculusCollege2 steps

  Find d/dx log_2(x)

  Derivatives

• Video lesson

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

  ▶1:09
  CalculusCollege7 steps

  Derivative of (ln(2x))/(sin(x²+1))

  Derivatives

• Video lesson

  $\frac{d}{dx}\left[\frac{{x}^{2}}{3}\right]$

  ▶0:45
  CalculusCollege3 steps

  Find d/dx(x²/3)

  Derivatives

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