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Derivative of exarctan⁡(x)e^{x} \arctan\left(x\right)

The derivative of exarctan⁡(x)e^{x} \arctan\left(x\right) is exarctan⁡(x)+ex11+x2e^{x} \arctan\left(x\right) + e^{x} \frac{1}{1 + x^{2}}.

f(x) =
ddx[exarctan⁡(x)]=exarctan⁡(x)+ex11+x2\frac{d}{dx}\left[e^{x} \arctan\left(x\right)\right] = e^{x} \arctan\left(x\right) + e^{x} \frac{1}{1 + x^{2}}

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Graph of f(x) = exp(x) * atan(x) and its derivative-3-2-1123-10-55101520slope 3.494f(x)f'(x)tangent at x = 1
Graph of f(x) = exp(x) * atan(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 3.4941.

How to work it out

  1. 1

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

    f(x)=exarctan⁡(x)f(x) = e^{x} \arctan\left(x\right)
  2. 2

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

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

    Differentiate each piece:

    u′=ex,v′=11+x2u' = e^{x},\quad v' = \frac{1}{1 + x^{2}}
  4. 4

    Put them in the rule:

    f′(x)=(ex)(arctan⁡(x))+(ex)(11+x2)f'(x) = \left(e^{x}\right)\left(\arctan\left(x\right)\right) + \left(e^{x}\right)\left(\frac{1}{1 + x^{2}}\right)
  5. 5

    Simplify:

    f′(x)=exarctan⁡(x)+ex11+x2f'(x) = e^{x} \arctan\left(x\right) + e^{x} \frac{1}{1 + x^{2}}
  6. 6

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

    f′′(x)=exarctan⁡(x)+ex11+x2+ex11+x2+ex−(2x)(1+x2)2f''(x) = e^{x} \arctan\left(x\right) + e^{x} \frac{1}{1 + x^{2}} + e^{x} \frac{1}{1 + x^{2}} + e^{x} \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[e^{x} \arctan\left(x\right)\right] = e^{x} \arctan\left(x\right) + e^{x} \frac{1}{1 + x^{2}} \]
\[ f(x) = e^{x} \arctan\left(x\right) \]
\[ u = e^{x},\quad v = \arctan\left(x\right) \]
\[ u' = e^{x},\quad v' = \frac{1}{1 + x^{2}} \]
\[ f'(x) = \left(e^{x}\right)\left(\arctan\left(x\right)\right) + \left(e^{x}\right)\left(\frac{1}{1 + x^{2}}\right) \]
\[ f'(x) = e^{x} \arctan\left(x\right) + e^{x} \frac{1}{1 + x^{2}} \]
\[ f''(x) = e^{x} \arctan\left(x\right) + e^{x} \frac{1}{1 + x^{2}} + e^{x} \frac{1}{1 + x^{2}} + e^{x} \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}} \]

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

xf(x)f'(x)
-2-0.14983629-0.12276923
-1-0.28893184-0.10499212
0.50.764425682.0834027
12.13493363.4940745
28.1807849.6585952
325.08775527.096309

Questions

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

The derivative of exarctan⁡(x)e^{x} \arctan\left(x\right) is exarctan⁡(x)+ex11+x2e^{x} \arctan\left(x\right) + e^{x} \frac{1}{1 + x^{2}}.

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

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

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

The product rule.

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

3.4940745.

Related derivatives

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