Questions: Replace ? with an expression that will make the equation valid. d/dx ln (x^8+7) = 1/(x^8+7) ? The missing expression is .

Solution Steps

The given equation is: \[ \frac{d}{d x} \ln \left(x^{8}+7\right)=\frac{1}{x^{8}+7} ? \] We need to find the expression that replaces the question mark to make the equation valid.

The derivative of \( \ln(u) \) with respect to \( x \) is \( \frac{1}{u} \cdot \frac{du}{dx} \), where \( u = x^{8} + 7 \). Applying the chain rule: \[ \frac{d}{d x} \ln \left(x^{8}+7\right) = \frac{1}{x^{8}+7} \cdot \frac{d}{d x} \left(x^{8}+7\right). \]

The derivative of \( x^{8} + 7 \) with respect to \( x \) is: \[ \frac{d}{d x} \left(x^{8}+7\right) = 8x^{7}. \]

Substituting the derivative into the chain rule result: \[ \frac{d}{d x} \ln \left(x^{8}+7\right) = \frac{1}{x^{8}+7} \cdot 8x^{7}. \]

Comparing this with the original equation: \[ \frac{d}{d x} \ln \left(x^{8}+7\right) = \frac{1}{x^{8}+7} \cdot 8x^{7}, \] the missing expression is \( 8x^{7} \).

Final Answer