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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 tan(xy) using proofs. To show this, we will use trigonometric identities along with the rules of differentiation. Different methods to prove this include: -</p>
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<p>We can derive the derivative of tan(xy) using proofs. To show this, we will use trigonometric identities along with the rules of differentiation. Different methods to prove this include: -</p>
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<ol><li>By First Principle </li>
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<ol><li>By First Principle </li>
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<li>Using Chain Rule </li>
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<li>Using Chain Rule </li>
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<li>Using Product Rule</li>
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<li>Using Product Rule</li>
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</ol><p>We will now demonstrate the differentiation of tan(xy) using the above methods:</p>
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</ol><p>We will now demonstrate the differentiation of tan(xy) using the above methods:</p>
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<h3>By First Principle</h3>
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<h3>By First Principle</h3>
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<p>The derivative of tan(xy) can be approached using the First Principle, expressing the derivative as the limit of the difference<a>quotient</a>. Consider f(x) = tan(xy). Its derivative can be expressed as: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h</p>
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<p>The derivative of tan(xy) can be approached using the First Principle, expressing the derivative as the limit of the difference<a>quotient</a>. Consider f(x) = tan(xy). Its derivative can be expressed as: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h</p>
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<p>Given f(x) = tan(xy), we substitute f(x + h) = tan((x + h)y).</p>
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<p>Given f(x) = tan(xy), we substitute f(x + h) = tan((x + h)y).</p>
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<p>Substituting these into the limit, we simplify using trigonometric identities and limits to obtain: f'(x) = sec²(xy) * (y + x * dy/dx).</p>
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<p>Substituting these into the limit, we simplify using trigonometric identities and limits to obtain: f'(x) = sec²(xy) * (y + x * dy/dx).</p>
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<p>Thus, the derivative is sec²(xy) * (y + x * dy/dx).</p>
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<p>Thus, the derivative is sec²(xy) * (y + x * dy/dx).</p>
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<h3>Using Chain Rule</h3>
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<h3>Using Chain Rule</h3>
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<p>To prove the differentiation of tan(xy) using the chain rule, we approach it as follows: Let u = xy, so tan(u) = sin(u)/cos(u).</p>
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<p>To prove the differentiation of tan(xy) using the chain rule, we approach it as follows: Let u = xy, so tan(u) = sin(u)/cos(u).</p>
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<p>Differentiating tan(u) with respect to u gives: d/dx (tan(xy)) = sec²(xy) * d/dx (xy).</p>
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<p>Differentiating tan(u) with respect to u gives: d/dx (tan(xy)) = sec²(xy) * d/dx (xy).</p>
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<p>Using the product rule on d/dx (xy), we get: d/dx (xy) = y + x * dy/dx.</p>
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<p>Using the product rule on d/dx (xy), we get: d/dx (xy) = y + x * dy/dx.</p>
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<p>Thus, d/dx (tan(xy)) = sec²(xy) * (y + x * dy/dx).</p>
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<p>Thus, d/dx (tan(xy)) = sec²(xy) * (y + x * dy/dx).</p>
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<h3>Using Product Rule</h3>
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<h3>Using Product Rule</h3>
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<p>We use the formula tan(xy) = sin(xy)/cos(xy) and apply the product rule: Let u = sin(xy) and v = cos(xy).</p>
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<p>We use the formula tan(xy) = sin(xy)/cos(xy) and apply the product rule: Let u = sin(xy) and v = cos(xy).</p>
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<p>Using the quotient rule: d/dx [u/v] = [v * d/dx(u) - u * d/dx(v)] / v².</p>
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<p>Using the quotient rule: d/dx [u/v] = [v * d/dx(u) - u * d/dx(v)] / v².</p>
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<p>Differentiating u and v using the chain and product rules, and substituting back,</p>
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<p>Differentiating u and v using the chain and product rules, and substituting back,</p>
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<p>we simplify to find: d/dx (tan(xy)) = sec²(xy) * (y + x * dy/dx).</p>
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<p>we simplify to find: d/dx (tan(xy)) = sec²(xy) * (y + x * dy/dx).</p>
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