Questions: [ fracdd x ln left(x^7+3right)=frac1x^7+3 ? ]

$[ fracdd x ln left(x^7+3right)=frac1x^7+3 ? ]$

Transcript text: \[ \frac{d}{d x} \ln \left(x^{7}+3\right)=\frac{1}{x^{7}+3} ? \]

Solution

Solution Steps

To solve this problem, we need to apply the chain rule for differentiation. The chain rule states that if you have a composite function $ f(g(x)) $, the derivative is $ f'(g(x)) \cdot g'(x) $. Here, $ f(u) = \ln(u) $ and $ g(x) = x^7 + 3 $. We find the derivative of each and multiply them.

Step 1: Identify the Functions

We have the expression $ \frac{d}{dx} \ln(x^7 + 3) $. Here, we identify the outer function as $ f(u) = \ln(u) $ and the inner function as $ g(x) = x^7 + 3 $.

Step 2: Compute the Derivatives

Using the chain rule, we find the derivative of the outer function: \[ f'(u) = \frac{1}{u} = \frac{1}{x^7 + 3} \] Next, we compute the derivative of the inner function: \[ g'(x) = \frac{d}{dx}(x^7 + 3) = 7x^6 \]

Step 3: Apply the Chain Rule

According to the chain rule, the derivative of the composite function is given by: \[ \frac{d}{dx} \ln(x^7 + 3) = f'(g(x)) \cdot g'(x) = \frac{1}{x^7 + 3} \cdot 7x^6 \] Thus, we can simplify this to: \[ \frac{7x^6}{x^7 + 3} \]

Final Answer

The missing expression that makes the equation valid is: \[ \boxed{7x^6} \]

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