Math Problem Statement

compute the derivative (with respect to x) of the following function

Solution

The function given is:

F(x)=0tanxdt1+t2.F(x) = \int_{0}^{\tan x} \frac{dt}{1 + t^2}.

To compute the derivative F(x)F'(x) with respect to xx, we can use the Fundamental Theorem of Calculus combined with the Chain Rule.

Step 1: Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if F(x)=a(x)b(x)f(t)dtF(x) = \int_{a(x)}^{b(x)} f(t) \, dt, then:

F(x)=f(b(x))b(x)f(a(x))a(x).F'(x) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x).

In this case, f(t)=11+t2f(t) = \frac{1}{1 + t^2}, a(x)=0a(x) = 0, and b(x)=tanxb(x) = \tan x.

Step 2: Applying the Theorem

Since a(x)=0a(x) = 0, a(x)=0a'(x) = 0, so the term involving a(x)a(x) will disappear. We only need to consider the upper limit b(x)=tanxb(x) = \tan x.

F(x)=11+(tanx)2ddx(tanx).F'(x) = \frac{1}{1 + (\tan x)^2} \cdot \frac{d}{dx}(\tan x).

Step 3: Simplification

The derivative of tanx\tan x with respect to xx is sec2x\sec^2 x. Therefore:

F(x)=11+tan2xsec2x.F'(x) = \frac{1}{1 + \tan^2 x} \cdot \sec^2 x.

Now, recall the trigonometric identity:

1+tan2x=sec2x.1 + \tan^2 x = \sec^2 x.

This simplifies the expression:

F(x)=sec2xsec2x=1.F'(x) = \frac{\sec^2 x}{\sec^2 x} = 1.

Final Answer

The derivative of F(x)F(x) with respect to xx is:

F(x)=1.F'(x) = 1.

Would you like to explore any more details or have any further questions?

Here are some related questions to consider:

  1. What if the upper limit was sinx\sin x instead of tanx\tan x?
  2. How would you differentiate F(x)=xtanxdt1+t2F(x) = \int_{x}^{\tan x} \frac{dt}{1 + t^2}?
  3. What is the derivative of ab(x)f(t)dt\int_{a}^{b(x)} f(t) \, dt with respect to xx when aa is a constant?
  4. How does the derivative change if the integrand is more complex, like dt1+t4\frac{dt}{1 + t^4}?
  5. What are some applications of the Fundamental Theorem of Calculus in physics?

Tip: The Fundamental Theorem of Calculus is powerful for evaluating integrals and their derivatives, especially when the limits of integration depend on the variable of differentiation.

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Math Problem Analysis

Mathematical Concepts

Calculus
Integral Calculus
Derivatives

Formulas

Fundamental Theorem of Calculus
Chain Rule
Trigonometric Identities

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Advanced High School