Question:
If \(g\) is a differentiable function of \(x\) such that \(g(x^{5})=3x^{7}\), for \(x\neq 0\), \(x\in \mathbb{R}\) then \(g'(3125)=\ldots\)
If \(g\) is a differentiable function of \(x\) such that \(g(x^{5})=3x^{7}\), for \(x\neq 0\), \(x\in \mathbb{R}\) then \(g'(3125)=\ldots\)
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Let u = x^5 so that g(u) = 3u^(7/5), then differentiate and substitute u = 3125.
Updated On: Nov 12, 2026
- \(125\)
- \(210\)
- \(105\)
- \(441\)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept
Put \(u=x^{5}\), so \(x=u^{1/5}\). Then the relation becomes a formula for g in terms of u.
Step 2: Find g
\(g(u)=3x^{7}=3u^{7/5}\).
Step 3: Differentiate
\(g'(u)=3\cdot\frac{7}{5}u^{2/5}=\frac{21}{5}u^{2/5}\).
Step 4: Substitute
\(3125=5^{5}\), so \(u^{2/5}=5^{2}=25\).
\(g'(3125)=\frac{21}{5}\times 25=105\).
Final Answer:
g'(3125) equals 105. \[ \boxed{\text{(C) }105} \]
Put \(u=x^{5}\), so \(x=u^{1/5}\). Then the relation becomes a formula for g in terms of u.
Step 2: Find g
\(g(u)=3x^{7}=3u^{7/5}\).
Step 3: Differentiate
\(g'(u)=3\cdot\frac{7}{5}u^{2/5}=\frac{21}{5}u^{2/5}\).
Step 4: Substitute
\(3125=5^{5}\), so \(u^{2/5}=5^{2}=25\).
\(g'(3125)=\frac{21}{5}\times 25=105\).
Final Answer:
g'(3125) equals 105. \[ \boxed{\text{(C) }105} \]
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