QQuestionMathematics
QuestionMathematics
Replace **?** with an expression that will make the equation valid.
\frac{d}{dx}(9x + 2)^5 = 5(9x + 2)^4 \quad ?
The missing expression is ☐.
Replace **?** with an expression that will make the equation valid.
\frac{d}{dx} e^{x^5 + 8} = e^{x^5 + 8} \quad ?
The missing expression is ☐.
Replace **?** with an expression that will make the equation valid.
\frac{d}{dx} \ln \left(x^5 + 8\right) = \frac{1}{x^5 + 8} \quad ?
The missing expression is ☐.
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Answer
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Step 1:: To make the first equation valid, we need to find the derivative of 1$ using the chain rule.
The chain rule states that the derivative of a composite function \(f(g(x))\) is \(\frac{df}{dx} = f'(g(x)) \cdot g'(x)\).
Step 2:: Let 1$.
Then, 1$.
Step 3:: Applying the chain rule, we get 1$.
Step 4:: Substitute 1$.
Step 5:: Simplify the equation: 1$.
Step 6:: Now, compare this result with the original equation 1$.
We can see that the original equation is missing a factor of 9.
Final Answer
--- Step 1: To make the second equation valid, we need to find the derivative of 1$.
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