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Derivative of exln⁡(x)e^{x} \ln\left(x\right)

The derivative of exln⁡(x)e^{x} \ln\left(x\right) is exln⁡(x)+ex1xe^{x} \ln\left(x\right) + e^{x} \frac{1}{x}.

f(x) =
ddx[exln⁡(x)]=exln⁡(x)+ex1x\frac{d}{dx}\left[e^{x} \ln\left(x\right)\right] = e^{x} \ln\left(x\right) + e^{x} \frac{1}{x}

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Graph of f(x) = exp(x) * ln(x) and its derivative-3-2-1123-10-5510152025slope 2.718f(x)f'(x)tangent at x = 1
Graph of f(x) = exp(x) * ln(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 2.7183.

How to work it out

  1. 1

    We differentiate f(x)=exln⁡(x)f(x) = e^{x} \ln\left(x\right) with respect to x. The rule that applies is the product rule.

    f(x)=exln⁡(x)f(x) = e^{x} \ln\left(x\right)
  2. 2

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. Let u=exu = e^{x} and v=ln⁡(x)v = \ln\left(x\right).

    u=ex,v=ln⁡(x)u = e^{x},\quad v = \ln\left(x\right)
  3. 3

    Differentiate each piece:

    u′=ex,v′=1xu' = e^{x},\quad v' = \frac{1}{x}
  4. 4

    Put them in the rule:

    f′(x)=(ex)(ln⁡(x))+(ex)(1x)f'(x) = \left(e^{x}\right)\left(\ln\left(x\right)\right) + \left(e^{x}\right)\left(\frac{1}{x}\right)
  5. 5

    Simplify:

    f′(x)=exln⁡(x)+ex1xf'(x) = e^{x} \ln\left(x\right) + e^{x} \frac{1}{x}
  6. 6

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

    f′′(x)=exln⁡(x)+ex1x+ex1x+ex−1x2f''(x) = e^{x} \ln\left(x\right) + e^{x} \frac{1}{x} + e^{x} \frac{1}{x} + e^{x} \frac{-1}{x^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[e^{x} \ln\left(x\right)\right] = e^{x} \ln\left(x\right) + e^{x} \frac{1}{x} \]
\[ f(x) = e^{x} \ln\left(x\right) \]
\[ u = e^{x},\quad v = \ln\left(x\right) \]
\[ u' = e^{x},\quad v' = \frac{1}{x} \]
\[ f'(x) = \left(e^{x}\right)\left(\ln\left(x\right)\right) + \left(e^{x}\right)\left(\frac{1}{x}\right) \]
\[ f'(x) = e^{x} \ln\left(x\right) + e^{x} \frac{1}{x} \]
\[ f''(x) = e^{x} \ln\left(x\right) + e^{x} \frac{1}{x} + e^{x} \frac{1}{x} + e^{x} \frac{-1}{x^{2}} \]

Values of f and f' for exp(x) * ln(x)

xf(x)f'(x)
0.5-1.14280652.154636
102.7182818
25.12170348.8162315
322.06621828.761397

Questions

What is the derivative of exp(x) * ln(x)?

The derivative of exln⁡(x)e^{x} \ln\left(x\right) is exln⁡(x)+ex1xe^{x} \ln\left(x\right) + e^{x} \frac{1}{x}.

What is the second derivative of exp(x) * ln(x)?

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

Which rule do you use for exp(x) * ln(x)?

The product rule.

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

2.7182818.

Related derivatives

    Derivative of exp(x) * ln(x): exp(x) * ln(x) + exp(x) * 1 / x (with steps) | WordGPT