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<p>Last updated on<strong>September 25, 2025</strong></p>
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<p>Last updated on<strong>September 25, 2025</strong></p>
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<p>In calculus, the chain rule is a fundamental formula for computing the derivative of the composition of two or more functions. It describes how to differentiate a composed function with respect to an independent variable. In this topic, we will learn the chain rule formula and its applications.</p>
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<p>In calculus, the chain rule is a fundamental formula for computing the derivative of the composition of two or more functions. It describes how to differentiate a composed function with respect to an independent variable. In this topic, we will learn the chain rule formula and its applications.</p>
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<h2>List of Math Formulas for the Chain Rule</h2>
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<h2>List of Math Formulas for the Chain Rule</h2>
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<p>The chain rule is used to differentiate composite<a>functions</a>. Let’s learn the<a>formula</a>and how to apply the chain rule for differentiation.</p>
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<p>The chain rule is used to differentiate composite<a>functions</a>. Let’s learn the<a>formula</a>and how to apply the chain rule for differentiation.</p>
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<h2>Math Formula for the Chain Rule</h2>
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<h2>Math Formula for the Chain Rule</h2>
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<p>The chain rule allows us to differentiate a composite function. If \( y = f(g(x))\) , then the derivative<a>of</a> y with respect to x is given by: \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\) </p>
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<p>The chain rule allows us to differentiate a composite function. If \( y = f(g(x))\) , then the derivative<a>of</a> y with respect to x is given by: \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\) </p>
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<p>This formula is essential for finding derivatives of complex functions.</p>
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<p>This formula is essential for finding derivatives of complex functions.</p>
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<h2>Application of the Chain Rule Formula</h2>
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<h2>Application of the Chain Rule Formula</h2>
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<p>The chain rule is applicable in various scenarios where functions are nested within each other.</p>
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<p>The chain rule is applicable in various scenarios where functions are nested within each other.</p>
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<p>Here’s a typical use case: If \(y = (3x^2 + 2)^5\) , let u = \(3x^2 + 2\) . Then \(y = u^5 \).</p>
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<p>Here’s a typical use case: If \(y = (3x^2 + 2)^5\) , let u = \(3x^2 + 2\) . Then \(y = u^5 \).</p>
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<p>Using the chain rule: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\) </p>
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<p>Using the chain rule: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\) </p>
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<p>\( \frac{dy}{du} = 5u^4 \quad \text{and} \quad \frac{du}{dx} = 6x \)</p>
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<p>\( \frac{dy}{du} = 5u^4 \quad \text{and} \quad \frac{du}{dx} = 6x \)</p>
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<p>Thus, \(\frac{dy}{dx} = 5(3x^2 + 2)^4 \cdot 6x\) .</p>
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<p>Thus, \(\frac{dy}{dx} = 5(3x^2 + 2)^4 \cdot 6x\) .</p>
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<h2>Importance of the Chain Rule Formula</h2>
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<h2>Importance of the Chain Rule Formula</h2>
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<p>In<a>calculus</a>and real-life applications, the chain rule formula is vital for analyzing and understanding the rates of change in composite functions. Here are some important points about the chain rule: </p>
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<p>In<a>calculus</a>and real-life applications, the chain rule formula is vital for analyzing and understanding the rates of change in composite functions. Here are some important points about the chain rule: </p>
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<p>The chain rule simplifies the process of finding derivatives of nested functions. </p>
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<p>The chain rule simplifies the process of finding derivatives of nested functions. </p>
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<p>It is essential for understanding complex systems, such as physics and engineering problems.</p>
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<p>It is essential for understanding complex systems, such as physics and engineering problems.</p>
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<h2>Tips and Tricks to Memorize the Chain Rule Formula</h2>
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<h2>Tips and Tricks to Memorize the Chain Rule Formula</h2>
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<p>Students often find calculus formulas tricky, but with some tips and tricks, mastering the chain rule becomes easier: - Remember the phrase:</p>
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<p>Students often find calculus formulas tricky, but with some tips and tricks, mastering the chain rule becomes easier: - Remember the phrase:</p>
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<p>"Derivative of the outside, times the derivative of the inside." </p>
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<p>"Derivative of the outside, times the derivative of the inside." </p>
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<p>Practice by breaking down complex functions into smaller parts. </p>
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<p>Practice by breaking down complex functions into smaller parts. </p>
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<p>Use visual aids like diagrams to understand how functions are composed.</p>
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<p>Use visual aids like diagrams to understand how functions are composed.</p>
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<h2>Real-Life Applications of the Chain Rule Formula</h2>
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<h2>Real-Life Applications of the Chain Rule Formula</h2>
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<p>In real life, the chain rule is crucial for understanding how changes in one quantity affect another. Here are some applications: </p>
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<p>In real life, the chain rule is crucial for understanding how changes in one quantity affect another. Here are some applications: </p>
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<p>In physics, calculating the velocity and acceleration of objects moving along a path defined by a composite function. </p>
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<p>In physics, calculating the velocity and acceleration of objects moving along a path defined by a composite function. </p>
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<p>In economics, modeling the<a>rate</a>of change of economic indicators affected by<a>multiple</a>underlying<a>factors</a>.</p>
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<p>In economics, modeling the<a>rate</a>of change of economic indicators affected by<a>multiple</a>underlying<a>factors</a>.</p>
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<h2>Common Mistakes and How to Avoid Them While Using the Chain Rule Formula</h2>
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<h2>Common Mistakes and How to Avoid Them While Using the Chain Rule Formula</h2>
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<p>Students often make errors when applying the chain rule. Here are some mistakes and ways to avoid them:</p>
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<p>Students often make errors when applying the chain rule. Here are some mistakes and ways to avoid them:</p>
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<h3>Problem 1</h3>
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<h3>Problem 1</h3>
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<p>Differentiate \( y = (2x^3 + 1)^4 \).</p>
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<p>Differentiate \( y = (2x^3 + 1)^4 \).</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The derivative is \(24x^2(2x^3 + 1)^3\) .</p>
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<p>The derivative is \(24x^2(2x^3 + 1)^3\) .</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Let \(u = 2x^3 + 1 \), then \( y = u^4\) .</p>
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<p>Let \(u = 2x^3 + 1 \), then \( y = u^4\) .</p>
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<p> \(\frac{dy}{du} = 4u^3 \quad \text{and} \quad \frac{du}{dx} = 6x^2\)</p>
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<p> \(\frac{dy}{du} = 4u^3 \quad \text{and} \quad \frac{du}{dx} = 6x^2\)</p>
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<p>Thus, \(\frac{dy}{dx} = 4(2x^3 + 1)^3 \cdot 6x^2 = 24x^2(2x^3 + 1)^3\) .</p>
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<p>Thus, \(\frac{dy}{dx} = 4(2x^3 + 1)^3 \cdot 6x^2 = 24x^2(2x^3 + 1)^3\) .</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 2</h3>
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<h3>Problem 2</h3>
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<p>If \( y = \sin(5x^2) \), find \( \frac{dy}{dx} \).</p>
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<p>If \( y = \sin(5x^2) \), find \( \frac{dy}{dx} \).</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The derivative is \(10x\cos(5x^2)\) .</p>
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<p>The derivative is \(10x\cos(5x^2)\) .</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Let \( u = 5x^2 \), then \(y = \sin(u)\) .</p>
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<p>Let \( u = 5x^2 \), then \(y = \sin(u)\) .</p>
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<p> \(\frac{dy}{du} = \cos(u) \quad \text{and} \quad \frac{du}{dx} = 10x \)</p>
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<p> \(\frac{dy}{du} = \cos(u) \quad \text{and} \quad \frac{du}{dx} = 10x \)</p>
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<p>Thus, \(\frac{dy}{dx} = \cos(5x^2) \cdot 10x = 10x\cos(5x^2)\) .</p>
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<p>Thus, \(\frac{dy}{dx} = \cos(5x^2) \cdot 10x = 10x\cos(5x^2)\) .</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h2>FAQs on the Chain Rule Formula</h2>
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<h2>FAQs on the Chain Rule Formula</h2>
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<h3>1.What is the chain rule formula?</h3>
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<h3>1.What is the chain rule formula?</h3>
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<p>The chain rule formula is used to differentiate composite functions: \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\) .</p>
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<p>The chain rule formula is used to differentiate composite functions: \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\) .</p>
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<h3>2.When do we use the chain rule?</h3>
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<h3>2.When do we use the chain rule?</h3>
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<p>The chain rule is used when differentiating a function that is the composition of two or more functions.</p>
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<p>The chain rule is used when differentiating a function that is the composition of two or more functions.</p>
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<h3>3.How do you identify the inner function in the chain rule?</h3>
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<h3>3.How do you identify the inner function in the chain rule?</h3>
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<p>The inner function is the function inside another function. Identify it by breaking down the composite function into simpler parts.</p>
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<p>The inner function is the function inside another function. Identify it by breaking down the composite function into simpler parts.</p>
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<h3>4.Can the chain rule be used repeatedly?</h3>
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<h3>4.Can the chain rule be used repeatedly?</h3>
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<p>Yes, the chain rule can be applied multiple times for functions composed of several nested functions.</p>
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<p>Yes, the chain rule can be applied multiple times for functions composed of several nested functions.</p>
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<h3>5.What if the function is not composite?</h3>
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<h3>5.What if the function is not composite?</h3>
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<p>For non-composite functions, the chain rule is not needed. Use basic differentiation rules instead.</p>
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<p>For non-composite functions, the chain rule is not needed. Use basic differentiation rules instead.</p>
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<h2>Glossary for the Chain Rule Formula</h2>
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<h2>Glossary for the Chain Rule Formula</h2>
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<ul><li><strong>Chain Rule:</strong>A formula used to differentiate composite functions by multiplying the derivative of the outer function by the derivative of the inner function.</li>
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<ul><li><strong>Chain Rule:</strong>A formula used to differentiate composite functions by multiplying the derivative of the outer function by the derivative of the inner function.</li>
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</ul><ul><li><strong>Composite Function:</strong>A function made up of two or more functions where the output of one function becomes the input of another.</li>
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</ul><ul><li><strong>Composite Function:</strong>A function made up of two or more functions where the output of one function becomes the input of another.</li>
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</ul><ul><li><strong>Derivative:</strong>A measure of how a function changes as its input changes.</li>
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</ul><ul><li><strong>Derivative:</strong>A measure of how a function changes as its input changes.</li>
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</ul><ul><li><strong>Inner Function:</strong>The function inside another function in a composite function.</li>
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</ul><ul><li><strong>Inner Function:</strong>The function inside another function in a composite function.</li>
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</ul><ul><li><strong>Outer Function:</strong>The external function applied to the result of the inner function in a composite function.</li>
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</ul><ul><li><strong>Outer Function:</strong>The external function applied to the result of the inner function in a composite function.</li>
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</ul><h2>Jaskaran Singh Saluja</h2>
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</ul><h2>Jaskaran Singh Saluja</h2>
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<h3>About the Author</h3>
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<h3>About the Author</h3>
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<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
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<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
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<h3>Fun Fact</h3>
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<h3>Fun Fact</h3>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>