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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Tutor-Verified Answer
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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