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Derivative of (e^(sin(x)))/(sqrt(x^2+x+1)): animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = ((-2x - 1) / 2 + sqrt(x² + x + 1)²·cos(x))·e^sin(x) / sqrt(x² + x + 1)³.

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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 (0.7 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Finding a derivative
$\frac{d}{dx}\left[\frac{{e}^{\sin\left( x\right)}}{\sqrt{{x}^{2}+ x+1}}\right]$
2. Steps, part 1
1. $\frac{d}{dx}\left[\frac{{e}^{\sin\left( x\right)}}{\sqrt{{x}^{2}+ x+1}}\right] = \frac{\left(\frac{-\left(2x\right)-1}{2}+{\sqrt{{x}^{2}+ x+1}}^{2} \cos\left( x\right)\right) {e}^{\sin\left( x\right)}}{{\sqrt{{x}^{2}+ x+1}}^{3}}$
  Quotient rule: (u/v)′ = (u′v − uv′)/v²
2. $\frac{d}{dx}\left[\sqrt{{x}^{2}+ x+1}\right] = \frac{\frac{2x+1}{2}}{\sqrt{{x}^{2}+ x+1}}$
  Chain rule — d/dx[√u] = u′/(2√u)
3. $\frac{d}{dx}\left[{x}^{2}+ x+1\right] = 2x+1$
  Sum rule: differentiate each term separately
4. $\frac{d}{dx}\left[{x}^{2}+ x\right] = 2x+1$
  Sum rule: differentiate each term separately
3. Steps, part 2
1. $\frac{d}{dx}\left[{x}^{2}\right] = 2x$
  Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
2. $\frac{d}{dx}\left[{e}^{\sin\left( x\right)}\right] = {e}^{\sin\left( x\right)} \cos\left( x\right)$
  Exponential rule: d/dx[eᵘ] = eᵘ·u′
3. $\frac{d}{dx}\left[\sin\left( x\right)\right] = \cos\left( x\right)$
  d/dx[sin x] = cos x
4. $f'(x) = \frac{\left(\frac{-\left(2x\right)-1}{2}+{\sqrt{{x}^{2}+ x+1}}^{2} \cos\left( x\right)\right) {e}^{\sin\left( x\right)}}{{\sqrt{{x}^{2}+ x+1}}^{3}}$
  Combine and simplify
4. The answer
$f'(x) = \frac{\left(\frac{-\left(2x\right)-1}{2}+{\sqrt{{x}^{2}+ x+1}}^{2} \cos\left( x\right)\right) {e}^{\sin\left( x\right)}}{{\sqrt{{x}^{2}+ x+1}}^{3}}$
f'(x) = ((-2x - 1) / 2 + sqrt(x² + x + 1)²·cos(x))·e^sin(x) / sqrt(x² + x + 1)³
5. Check
- Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

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

Steps, part 1. Quotient rule: (u/v) prime equals (u prime v minus uv prime )/v squared. Chain rule — d by d x[the square root of u] equals u prime /(2the square root of u. Sum rule: differentiate each term separately. Sum rule: differentiate each term separately.

Steps, part 2. Power rule: d by d x[x to the power n] equals n times x to the power n minus 1. Exponential rule: d by d x[eᵘ] equals eᵘ times u prime. d by d x[sin x] equals cos x. Combine and simplify.

The answer. So the answer is f prime of x equals ((negative 2x minus 1) / 2 plus sqrt(x squared plus x plus 1) squared times cos(x)) times e to the power sin(x) / sqrt(x squared plus x plus 1) cubed. 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 (e^(sin(x)))/(sqrt(x^2+x+1)): Animated Lesson