Given The Function $f(x)=\int_0^x \frac{1}{1+t^2} , Dt$, Find $f^{\prime}(x)$.(A) $f {\prime}(x)=\frac{1}{1+x 4}$(B) $ F ′ ( X ) = 2 X 1 + X 4 F^{\prime}(x)=\frac{2x}{1+x^4} F ′ ( X ) = 1 + X 4 2 X [/tex](C) $f^{\prime}(x)=\arctan X^2$(D)

Mar 12, 2025 by ADMIN 249 views

**Finding the Derivative of a Definite Integral**

Introduction

In calculus, the derivative of a function is a measure of how the function changes as its input changes. One of the fundamental theorems in calculus is the Fundamental Theorem of Calculus (FTC), which relates the derivative of a function to the definite integral of that function. In this article, we will explore how to find the derivative of a definite integral using the FTC.

The Fundamental Theorem of Calculus

The FTC states that if we have a function $f(x)$ and we define a new function $F(x)$ as the definite integral of $f(x)$ from $a$ to $x$ , then the derivative of $F(x)$ is equal to $f(x)$ . Mathematically, this can be expressed as:

F^{\prime}(x) = f(x)

where $F(x) = \int_{a}^{x} f(t) \, dt$

Applying the Fundamental Theorem of Calculus

Now, let's apply the FTC to the given function:

f(x) = \int_{0}^{x} \frac{1}{1+t^2} \, dt

We want to find the derivative of $f(x)$ , which is denoted as $f^{\prime}(x)$ . Using the FTC, we can write:

f^{\prime}(x) = \frac{d}{dx} \int_{0}^{x} \frac{1}{1+t^2} \, dt

Using the Chain Rule

To find the derivative of the definite integral, we can use the chain rule. The chain rule states that if we have a composite function of the form $h(g(x))$ , then the derivative of $h(g(x))$ is equal to the derivative of $h$ evaluated at $g(x)$ , multiplied by the derivative of $g(x)$ . In this case, we can rewrite the definite integral as a composite function:

f(x) = \int_{0}^{x} \frac{1}{1+t^2} \, dt = \int_{0}^{x} \frac{1}{1+(t^2)^1} \, dt

Now, we can apply the chain rule to find the derivative of $f(x)$ :

f^{\prime}(x) = \frac{d}{dx} \int_{0}^{x} \frac{1}{1+(t^2)^1} \, dt = \frac{d}{dx} \int_{0}^{x} \frac{1}{1+(t^2)^{1/2}} \, dt

Finding the Derivative of the Integrand

To find the derivative of the integrand, we can use the power rule of differentiation. The power rule states that if we have a function of the form $x^n$ , then the derivative of $x^n$ is equal to $n \cdot x^{n-1}$ . In this case, we can rewrite the integrand as:

\frac{1}{1+t^2} = \frac{1}{1+(t^2)^{1/2}}

Now, we can apply the power rule to find the derivative of the integrand:

\frac{d}{dt} \frac{1}{1+t^2} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{d}{dt} \frac{1}{1+(t^2)^{1/2}} = \frac{<br/> **Q&A: Finding the Derivative of a Definite Integral** =====================================================

Q: What is the Fundamental Theorem of Calculus?

A: The Fundamental Theorem of Calculus (FTC) is a fundamental theorem in calculus that relates the derivative of a function to the definite integral of that function. It states that if we have a function $f(x)$ and we define a new function $F(x)$ as the definite integral of $f(x)$ from $a$ to $x$ , then the derivative of $F(x)$ is equal to $f(x)$ .

Q: How do I apply the Fundamental Theorem of Calculus to find the derivative of a definite integral?

A: To apply the FTC, you need to follow these steps:

Define the function $f(x)$ and the definite integral $F(x) = \int_{a}^{x} f(t) \, dt$
Use the FTC to write the derivative of $F(x)$ as $F^{\prime}(x) = f(x)$
Use the chain rule to find the derivative of the integrand
Simplify the expression to find the final answer

Q: What is the chain rule?

A: The chain rule is a rule in calculus that allows us to find the derivative of a composite function. It states that if we have a composite function of the form $h(g(x))$ , then the derivative of $h(g(x))$ is equal to the derivative of $h$ evaluated at $g(x)$ , multiplied by the derivative of $g(x)$ .

Q: How do I use the chain rule to find the derivative of a definite integral?

A: To use the chain rule, you need to follow these steps:

Rewrite the definite integral as a composite function
Apply the chain rule to find the derivative of the composite function
Simplify the expression to find the final answer

Q: What is the power rule of differentiation?

A: The power rule of differentiation is a rule in calculus that allows us to find the derivative of a function of the form $x^n$ . It states that the derivative of $x^n$ is equal to $n \cdot x^{n-1}$ .

Q: How do I use the power rule to find the derivative of a definite integral?

A: To use the power rule, you need to follow these steps:

Rewrite the integrand as a function of the form $x^n$
Apply the power rule to find the derivative of the integrand
Simplify the expression to find the final answer

Q: What is the final answer to the problem?

A: The final answer to the problem is:

f^{\prime}(x) = \frac{2x}{1+x^4} </span></p> <p>This is the derivative of the definite integral <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>x</mi></msubsup><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mtext> </mtext><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">f(x) = \int_{0}^{x} \frac{1}{1+t^2} \, dt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2626em;vertical-align:-0.4033em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8593em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3558em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span>.</p> <h2><strong>Conclusion</strong></h2> <p>In this article, we have discussed how to find the derivative of a definite integral using the Fundamental Theorem of Calculus and the chain rule. We have also discussed how to use the power rule of differentiation to find the derivative of a function of the form <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">x^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span>. We have applied these concepts to find the derivative of the definite integral <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>x</mi></msubsup><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mtext> </mtext><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">f(x) = \int_{0}^{x} \frac{1}{1+t^2} \, dt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2626em;vertical-align:-0.4033em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8593em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3558em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span> and have obtained the final answer.</p> <h2><strong>References</strong></h2> <ul> <li>[1] &quot;Calculus&quot; by Michael Spivak</li> <li>[2] &quot;Calculus&quot; by James Stewart</li> <li>[3] &quot;The Fundamental Theorem of Calculus&quot; by Wolfram MathWorld</li> </ul> <h2><strong>Glossary</strong></h2> <ul> <li><strong>Definite integral</strong>: A type of integral that has a specific upper and lower bound.</li> <li><strong>Fundamental Theorem of Calculus</strong>: A theorem in calculus that relates the derivative of a function to the definite integral of that function.</li> <li><strong>Chain rule</strong>: A rule in calculus that allows us to find the derivative of a composite function.</li> <li><strong>Power rule of differentiation</strong>: A rule in calculus that allows us to find the derivative of a function of the form <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">x^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span>.</li> </ul>