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2026-01-01
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2026-02-28
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<p>We can derive the derivative of sin(x²) using proofs. To show this, we will use trigonometric identities along with the rules of differentiation.</p>
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<p>We can derive the derivative of sin(x²) using proofs. To show this, we will use trigonometric identities along with the rules of differentiation.</p>
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<p>There are several methods we use to prove this, such as:</p>
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<p>There are several methods we use to prove this, such as:</p>
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<p>By First Principle</p>
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<p>By First Principle</p>
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<p>Using Chain Rule</p>
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<p>Using Chain Rule</p>
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<p>Using Product Rule</p>
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<p>Using Product Rule</p>
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<p>We will now demonstrate that the differentiation of sin(x²) results in 2x cos(x²) using the above-mentioned methods:</p>
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<p>We will now demonstrate that the differentiation of sin(x²) results in 2x cos(x²) using the above-mentioned methods:</p>
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<p>Using Chain Rule</p>
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<p>Using Chain Rule</p>
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<p>To prove the differentiation of sin(x²) using the chain rule, We use the formula: y = sin(u), where u = x² Then dy/du = cos(u) and du/dx = 2x</p>
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<p>To prove the differentiation of sin(x²) using the chain rule, We use the formula: y = sin(u), where u = x² Then dy/du = cos(u) and du/dx = 2x</p>
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<p>Using the chain rule: dy/dx = dy/du * du/dx dy/dx = cos(x²) * 2x</p>
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<p>Using the chain rule: dy/dx = dy/du * du/dx dy/dx = cos(x²) * 2x</p>
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<p>Thus, d/dx (sin(x²)) = 2x cos(x²).</p>
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<p>Thus, d/dx (sin(x²)) = 2x cos(x²).</p>
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<p>Using First Principle The derivative of sin(x²) can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>.</p>
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<p>Using First Principle The derivative of sin(x²) can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>.</p>
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<p>To find the derivative of sin(x²) using the first principle, we will consider f(x) = sin(x²).</p>
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<p>To find the derivative of sin(x²) using the first principle, we will consider f(x) = sin(x²).</p>
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<p>Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)</p>
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<p>Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)</p>
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<p>Given that f(x) = sin(x²), we write f(x + h) = sin((x + h)²).</p>
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<p>Given that f(x) = sin(x²), we write f(x + h) = sin((x + h)²).</p>
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<p>Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [sin((x + h)²) - sin(x²)] / h</p>
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<p>Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [sin((x + h)²) - sin(x²)] / h</p>
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<p>Using trigonometric identities and limit properties, and simplifying, f'(x) = 2x cos(x²).</p>
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<p>Using trigonometric identities and limit properties, and simplifying, f'(x) = 2x cos(x²).</p>
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<p>Using Product Rule</p>
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<p>Using Product Rule</p>
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<p>The<a>product</a>rule is not directly applicable to sin(x²) as it is not a product of functions. But for functions involving<a>multiplication</a>of derivatives, the product rule can be useful.</p>
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<p>The<a>product</a>rule is not directly applicable to sin(x²) as it is not a product of functions. But for functions involving<a>multiplication</a>of derivatives, the product rule can be useful.</p>
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