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Derivative of ex2+1e^{x^{2} + 1}

The derivative of ex2+1e^{x^{2} + 1} is ex2+12xe^{x^{2} + 1} 2 x.

f(x) =
ddx[ex2+1]=ex2+12x\frac{d}{dx}\left[e^{x^{2} + 1}\right] = e^{x^{2} + 1} 2 x

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Graph of f(x) = exp(x^2 + 1) and its derivative-3-2-1123-500050001000015000slope 14.78f(x)f'(x)tangent at x = 1
Graph of f(x) = exp(x^2 + 1) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 14.778.

How to work it out

  1. 1

    We differentiate f(x)=ex2+1f(x) = e^{x^{2} + 1} with respect to x. The rule that applies is the chain rule.

    f(x)=ex2+1f(x) = e^{x^{2} + 1}
  2. 2

    Chain rule: differentiate the outer function, keep the inside, then multiply by the derivative of the inside. The inside is g(x)=x2+1g(x) = x^{2} + 1.

    g(x)=x2+1,g′(x)=2xg(x) = x^{2} + 1,\quad g'(x) = 2 x
  3. 3

    The outer function is eue^{u}; its derivative evaluated at g(x), times g'(x):

    f′(x)=[eu]u=g(x)⋅g′(x)f'(x) = \left[e^{u}\right]_{u=g(x)} \cdot g'(x)
  4. 4

    Simplify:

    f′(x)=ex2+12xf'(x) = e^{x^{2} + 1} 2 x
  5. 5

    Differentiating once more gives the second derivative, which tells you about curvature:

    f′′(x)=ex2+12x2x+2ex2+1f''(x) = e^{x^{2} + 1} 2 x 2 x + 2 e^{x^{2} + 1}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[e^{x^{2} + 1}\right] = e^{x^{2} + 1} 2 x \]
\[ f(x) = e^{x^{2} + 1} \]
\[ g(x) = x^{2} + 1,\quad g'(x) = 2 x \]
\[ f'(x) = \left[e^{u}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = e^{x^{2} + 1} 2 x \]
\[ f''(x) = e^{x^{2} + 1} 2 x 2 x + 2 e^{x^{2} + 1} \]

Values of f and f' for exp(x2x^{2} + 1)

xf(x)f'(x)
-2148.41316-593.65264
-17.3890561-14.778112
0.53.4903433.490343
17.389056114.778112
2148.41316593.65264
322026.466132158.79

Questions

What is the derivative of exp(x2x^{2} + 1)?

The derivative of ex2+1e^{x^{2} + 1} is ex2+12xe^{x^{2} + 1} 2 x.

What is the second derivative of exp(x2x^{2} + 1)?

exp(x2x^{2} + 1) * 2x * 2x + 2exp(x2x^{2} + 1).

Which rule do you use for exp(x2x^{2} + 1)?

The chain rule.

What is the slope of exp(x2x^{2} + 1) at x = 1?

14.778112.

Related derivatives

    Derivative of exp(x^2 + 1): exp(x^2 + 1) * 2x (with steps) | WordGPT