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Derivative of e3xe^{3 x}

The derivative of e3xe^{3 x} is 3e3x3 e^{3 x}.

f(x) =
ddx[e3x]=3e3x\frac{d}{dx}\left[e^{3 x}\right] = 3 e^{3 x}

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

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Graph of f(x) = exp(3x) and its derivative-3-2-1123-10001000200030004000500060007000slope 60.26f(x)f'(x)tangent at x = 1
Graph of f(x) = exp(3x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 60.257.

How to work it out

  1. 1

    We differentiate f(x)=e3xf(x) = e^{3 x} with respect to x. The rule that applies is the chain rule.

    f(x)=e3xf(x) = e^{3 x}
  2. 2

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

    g(x)=3x,g′(x)=3g(x) = 3 x,\quad g'(x) = 3
  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)=3e3xf'(x) = 3 e^{3 x}
  5. 5

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

    f′′(x)=9e3xf''(x) = 9 e^{3 x}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[e^{3 x}\right] = 3 e^{3 x} \]
\[ f(x) = e^{3 x} \]
\[ g(x) = 3 x,\quad g'(x) = 3 \]
\[ f'(x) = \left[e^{u}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = 3 e^{3 x} \]
\[ f''(x) = 9 e^{3 x} \]

Values of f and f' for exp(3x)

xf(x)f'(x)
-20.00247875220.0074362565
-10.0497870680.14936121
0.54.481689113.445067
120.08553760.256611
2403.428791210.2864
38103.083924309.252

Questions

What is the derivative of exp(3x)?

The derivative of e3xe^{3 x} is 3e3x3 e^{3 x}.

What is the second derivative of exp(3x)?

9exp(3x).

Which rule do you use for exp(3x)?

The chain rule.

What is the slope of exp(3x) at x = 1?

60.256611.

Related derivatives

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