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Simplify the expression $x^2+x+\frac{1}{4}$

Step-by-step Solution

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Solution

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  • Write in simplest form
  • Solve by quadratic formula (general formula)
  • Find the derivative using the definition
  • Simplify
  • Find the integral
  • Find the derivative
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Limit of this function

$\lim_{x\to0}\left(x^2+x+\frac{1}{4}\right)=\frac{1}{4}$
See step-by-step solution

Derivative of this function

$\frac{d}{dx}\left(x^2+x+\frac{1}{4}\right)=2x+1$
See step-by-step solution

Integral of this function

$\int\left(x^2+x+\frac{1}{4}\right)dx=\frac{x^{3}}{3}+\frac{1}{2}x^2+\frac{1}{4}x+C_0$
See step-by-step solution

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Main Topic: Implicit Differentiation

Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions. For differentiating an implicit function y(x), defined by an equation R(x, y) = 0, it is not generally possible to solve it explicitly for y(x) and then differentiate. Instead, one can differentiate R(x, y) with respect to x and y and then solve a linear equation in dy/dx for getting explicitly the derivative in terms of x and y.

Used Formulas

See formulas (4)

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