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DerivativesCalculusCollegeabout 1:26

Derivative of (sqrt(e^(x)+1))(cos(sin(x))): animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = (e^x·cos(sin(x)) / 2 - sqrt(e^x + 1)²·sin(sin(x))·cos(x)) / sqrt(e^x + 1).

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Transcript

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.6 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Finding a derivative
$\frac{d}{dx}\left[\sqrt{{e}^{x}+1} \cos\left(\sin\left( x\right)\right)\right]$
2. Steps, part 1
1. $\frac{d}{dx}\left[\sqrt{{e}^{x}+1} \cos\left(\sin\left( x\right)\right)\right] = \frac{\frac{{e}^{x} \cos\left(\sin\left( x\right)\right)}{2}-{\sqrt{{e}^{x}+1}}^{2} \sin\left(\sin\left( x\right)\right) \cos\left( x\right)}{\sqrt{{e}^{x}+1}}$
  Product rule: (uv)′ = u′v + uv′
2. $\frac{d}{dx}\left[\cos\left(\sin\left( x\right)\right)\right] = -\left(\sin\left(\sin\left( x\right)\right) \cos\left( x\right)\right)$
  Chain rule — d/dx[cos u] = −sin u · u′
3. $\frac{d}{dx}\left[\sin\left( x\right)\right] = \cos\left( x\right)$
  d/dx[sin x] = cos x
4. $\frac{d}{dx}\left[\sqrt{{e}^{x}+1}\right] = \frac{\frac{{e}^{x}}{2}}{\sqrt{{e}^{x}+1}}$
  Chain rule — d/dx[√u] = u′/(2√u)
3. Steps, part 2
1. $\frac{d}{dx}\left[{e}^{x}+1\right] = {e}^{x}$
  Sum rule: differentiate each term separately
2. $\frac{d}{dx}\left[{e}^{x}\right] = {e}^{x}$
  Exponential rule: d/dx[eᵘ] = eᵘ·u′
3. $f'(x) = \frac{\frac{{e}^{x} \cos\left(\sin\left( x\right)\right)}{2}-{\sqrt{{e}^{x}+1}}^{2} \sin\left(\sin\left( x\right)\right) \cos\left( x\right)}{\sqrt{{e}^{x}+1}}$
  Combine and simplify
4. The answer
$f'(x) = \frac{\frac{{e}^{x} \cos\left(\sin\left( x\right)\right)}{2}-{\sqrt{{e}^{x}+1}}^{2} \sin\left(\sin\left( x\right)\right) \cos\left( x\right)}{\sqrt{{e}^{x}+1}}$
f'(x) = (e^x·cos(sin(x)) / 2 - sqrt(e^x + 1)²·sin(sin(x))·cos(x)) / sqrt(e^x + 1)
5. Check
- Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

The problem. Here is the problem: derivative of (sqrt(e to the power x plus 1))(cos(sin(x))). Find the derivative of (sqrt(e to the power x plus 1)) (cos(sin(x))) with respect to x.

Steps, part 1. Product rule: (uv) prime equals u prime v plus uv prime. Chain rule — d by d x[cos u] equals negative sin u times u prime. d by d x[sin x] equals cos x. Chain rule — d by d x[the square root of u] equals u prime /(2the square root of u.

Steps, part 2. Sum rule: differentiate each term separately. Exponential rule: d by d x[eᵘ] equals eᵘ times u prime. Combine and simplify.

The answer. So the answer is f prime of x equals (e to the power x times cos(sin(x)) / 2 minus sqrt(e to the power x plus 1) squared times sin(sin(x)) times cos(x)) / sqrt(e to the power x plus 1). The derivative is built from the outside in: sums split into terms, constants factor out, products and quotients use their rules, and every composite function multiplies by the derivative of its inside (chain rule).

Check. Cross-checked numerically against an independent symbolic differentiator at 8 sample points, which checks out.

About derivatives

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

Derivative of (sqrt(e^(x)+1))(cos(sin(x))): Animated Lesson