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Derivative of x2tan⁡(x)x^{2} \tan\left(x\right)

The derivative of x2tan⁡(x)x^{2} \tan\left(x\right) is 2xtan⁡(x)+x21cos⁡(x)22 x \tan\left(x\right) + x^{2} \frac{1}{\cos\left(x\right)^{2}}.

f(x) =
ddx[x2tan⁡(x)]=2xtan⁡(x)+x21cos⁡(x)2\frac{d}{dx}\left[x^{2} \tan\left(x\right)\right] = 2 x \tan\left(x\right) + x^{2} \frac{1}{\cos\left(x\right)^{2}}

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Graph of f(x) = x^2 * tan(x) and its derivative-3-2-112350100150slope 6.54f(x)f'(x)tangent at x = 1
Graph of f(x) = x^2 * tan(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 6.5403.

How to work it out

  1. 1

    We differentiate f(x)=x2tan⁡(x)f(x) = x^{2} \tan\left(x\right) with respect to x. The rule that applies is the product rule.

    f(x)=x2tan⁡(x)f(x) = x^{2} \tan\left(x\right)
  2. 2

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. Let u=x2u = x^{2} and v=tan⁡(x)v = \tan\left(x\right).

    u=x2,v=tan⁡(x)u = x^{2},\quad v = \tan\left(x\right)
  3. 3

    Differentiate each piece:

    u′=2x,v′=1cos⁡(x)2u' = 2 x,\quad v' = \frac{1}{\cos\left(x\right)^{2}}
  4. 4

    Put them in the rule:

    f′(x)=(2x)(tan⁡(x))+(x2)(1cos⁡(x)2)f'(x) = \left(2 x\right)\left(\tan\left(x\right)\right) + \left(x^{2}\right)\left(\frac{1}{\cos\left(x\right)^{2}}\right)
  5. 5

    Simplify:

    f′(x)=2xtan⁡(x)+x21cos⁡(x)2f'(x) = 2 x \tan\left(x\right) + x^{2} \frac{1}{\cos\left(x\right)^{2}}
  6. 6

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

    f′′(x)=2tan⁡(x)+2x1cos⁡(x)2+2x1cos⁡(x)2+x2−(2cos⁡(x)−sin⁡(x))(cos⁡(x)2)2f''(x) = 2 \tan\left(x\right) + 2 x \frac{1}{\cos\left(x\right)^{2}} + 2 x \frac{1}{\cos\left(x\right)^{2}} + x^{2} \frac{-\left(2 \cos\left(x\right) -\sin\left(x\right)\right)}{\left(\cos\left(x\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[x^{2} \tan\left(x\right)\right] = 2 x \tan\left(x\right) + x^{2} \frac{1}{\cos\left(x\right)^{2}} \]
\[ f(x) = x^{2} \tan\left(x\right) \]
\[ u = x^{2},\quad v = \tan\left(x\right) \]
\[ u' = 2 x,\quad v' = \frac{1}{\cos\left(x\right)^{2}} \]
\[ f'(x) = \left(2 x\right)\left(\tan\left(x\right)\right) + \left(x^{2}\right)\left(\frac{1}{\cos\left(x\right)^{2}}\right) \]
\[ f'(x) = 2 x \tan\left(x\right) + x^{2} \frac{1}{\cos\left(x\right)^{2}} \]
\[ f''(x) = 2 \tan\left(x\right) + 2 x \frac{1}{\cos\left(x\right)^{2}} + 2 x \frac{1}{\cos\left(x\right)^{2}} + x^{2} \frac{-\left(2 \cos\left(x\right) -\sin\left(x\right)\right)}{\left(\cos\left(x\right)^{2}\right)^{2}} \]

Values of f and f' for x2x^{2} * tan(x)

xf(x)f'(x)
-28.740159514.357437
-1-1.55740776.5403343
0.50.136575620.87091409
11.55740776.5403343
2-8.740159514.357437
3-1.28291898.3275964

Questions

What is the derivative of x2x^{2} * tan(x)?

The derivative of x2tan⁡(x)x^{2} \tan\left(x\right) is 2xtan⁡(x)+x21cos⁡(x)22 x \tan\left(x\right) + x^{2} \frac{1}{\cos\left(x\right)^{2}}.

What is the second derivative of x2x^{2} * tan(x)?

2tan(x) + 2x * 1 / cos(x)^2 + 2x * 1 / cos(x)^2 + x2x^{2} * -(2cos(x) * -sin(x)) / (cos(x)^2)^2.

Which rule do you use for x2x^{2} * tan(x)?

The product rule.

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

6.5403343.

Related derivatives

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