Ray Burns
01/01/2024 · Primary School

¿Cuál es el valor de \( \frac{d}{d x}\left(-\frac{1}{x^{4}}-\frac{5}{x^{2}}\right) \) en \( x=2 \) ?

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Step-by-step Solution

Find the first order derivative with respect to \( x \) for \( -\frac{1}{x^{4}}-\frac{5}{x^{2}} \). Evaluate the derivative by following steps: - step0: Evaluate the derivative: \(\frac{d}{dx}\left(-\frac{1}{x^{4}}-\frac{5}{x^{2}}\right)\) - step1: Subtract the terms: \(\frac{d}{dx}\left(-\frac{1+5x^{2}}{x^{4}}\right)\) - step2: Use differentiation rules: \(-\frac{\frac{d}{dx}\left(1+5x^{2}\right)\times x^{4}-\left(1+5x^{2}\right)\times \frac{d}{dx}\left(x^{4}\right)}{\left(x^{4}\right)^{2}}\) - step3: Calculate: \(-\frac{10x\times x^{4}-\left(1+5x^{2}\right)\times 4x^{3}}{\left(x^{4}\right)^{2}}\) - step4: Calculate: \(-\frac{10x^{5}-4x^{3}\left(1+5x^{2}\right)}{x^{8}}\) - step5: Calculate: \(-\frac{-10x^{5}-4x^{3}}{x^{8}}\) - step6: Calculate: \(\frac{10x^{2}+4}{x^{5}}\) Substitute \( x=2 \) into the expression \( \frac{10x^{2}+4}{x^{5}} \). Evaluate the expression by following steps: - step0: Evaluate: \(\frac{10x^{2}+4}{x^{5}}\) - step1: Substitute: \(\frac{10\times 2^{2}+4}{2^{5}}\) - step2: Multiply the terms: \(\frac{40+4}{2^{5}}\) - step3: Add the numbers: \(\frac{44}{2^{5}}\) - step4: Rewrite the expression: \(\frac{4\times 11}{2^{5}}\) - step5: Rewrite the expression: \(\frac{2^{2}\times 11}{2^{5}}\) - step6: Reduce the fraction: \(\frac{11}{2^{3}}\) - step7: Simplify: \(\frac{11}{8}\) El valor de \( \frac{d}{d x}\left(-\frac{1}{x^{4}}-\frac{5}{x^{2}}\right) \) en \( x=2 \) es \( \frac{11}{8} \) o \( 1\frac{3}{8} \) o \( 1.375 \).

Quick Answer

El valor de \( \frac{d}{d x}\left(-\frac{1}{x^{4}}-\frac{5}{x^{2}}\right) \) en \( x=2 \) es \( \frac{11}{8} \).
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