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Differentiating an Integral with Variable Limits (Leibniz Rule Explained Step-by-Step)

  • Writer: Tyler Buffone
    Tyler Buffone
  • Dec 9, 2024
  • 1 min read

Updated: Oct 9

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Example


Consider the following function:


Mathematical equation on a dark background: F(x) = ∫ [(x²+1)/(2+cosx)] (1/t) dt. White handwritten text in a neat style.

Find its derivative.


Solution


Use the Leibniz Integral rule:


Mathematical equation on a dark background reads: "If F(x) = ∫[a(x) to b(x)] F(t)dt, then F'(x) = F(b(x))b'(x) - F(a(x))a'(x)."

Remark: this is also known as the Differentiation Under the Integral Sign Formula.


In this case:


f(t) = (1 / t), a(x) = 2 + cos(x), and b(x) = x² + 1.


So:


  • f(a(x)) = (1 / (2 + cos(x))

  • a'(x) = -sin(x)

  • f(b(x)) = (1 / (x² + 1))

  • b'(x) = 2x


Hence, the derivative of F(x) is:


Math equation showing F'(x) with derivatives involving fractions and trigonometric functions on a dark background.

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