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

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.

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 \( g(x) = x^7 + 3 \).

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: \[ g'(x) = \frac{d}{dx}(x^7 + 3) = 7x^6 \]

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