WordGPT

Derivative of ex3e^{x^{3}}

The derivative of ex3e^{x^{3}} is ex33x2e^{x^{3}} 3 x^{2}.

f(x) =
ddx[ex3]=ex33x2\frac{d}{dx}\left[e^{x^{3}}\right] = e^{x^{3}} 3 x^{2}

Type any function of x; the derivative is computed symbolically in your browser.

Explain it step by step in chatMake a short video of it
Graph of f(x) = exp(x^3) and its derivative-3-2-11231e+92e+93e+94e+9slope 8.155f(x)f'(x)tangent at x = 1
Graph of f(x) = exp(x^3) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 8.1548.

How to work it out

  1. 1

    We differentiate f(x)=ex3f(x) = e^{x^{3}} with respect to x. The rule that applies is the chain rule.

    f(x)=ex3f(x) = e^{x^{3}}
  2. 2

    Chain rule: differentiate the outer function, keep the inside, then multiply by the derivative of the inside. The inside is g(x)=x3g(x) = x^{3}.

    g(x)=x3,g′(x)=3x2g(x) = x^{3},\quad g'(x) = 3 x^{2}
  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)=ex33x2f'(x) = e^{x^{3}} 3 x^{2}
  5. 5

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

    f′′(x)=ex33x23x2+ex36xf''(x) = e^{x^{3}} 3 x^{2} 3 x^{2} + e^{x^{3}} 6 x
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[e^{x^{3}}\right] = e^{x^{3}} 3 x^{2} \]
\[ f(x) = e^{x^{3}} \]
\[ g(x) = x^{3},\quad g'(x) = 3 x^{2} \]
\[ f'(x) = \left[e^{u}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = e^{x^{3}} 3 x^{2} \]
\[ f''(x) = e^{x^{3}} 3 x^{2} 3 x^{2} + e^{x^{3}} 6 x \]

Values of f and f' for exp(x3x^{3})

xf(x)f'(x)
-20.000335462630.0040255515
-10.367879441.1036383
0.51.13314850.84986134
12.71828188.1548455
22980.95835771.496
353204824000014365302000000

Questions

What is the derivative of exp(x3x^{3})?

The derivative of ex3e^{x^{3}} is ex33x2e^{x^{3}} 3 x^{2}.

What is the second derivative of exp(x3x^{3})?

exp(x3x^{3}) * 3 * x2x^{2} * 3 * x2x^{2} + exp(x3x^{3}) * 6x.

Which rule do you use for exp(x3x^{3})?

The chain rule.

What is the slope of exp(x3x^{3}) at x = 1?

8.1548455.

Related derivatives

    Derivative of exp(x^3): exp(x^3) * 3 * x^2 (with steps) | WordGPT