Find the derivative of e^x sin(x): animated lesson

1. 1. The problem
   Finding a derivative

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

   f'(x) = e^x·(sin(x) + cos(x))

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.

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