Questions: Find f'(x) if f(x) = sqrt(3+3x).

Solution Steps

To find the derivative \( f^{\prime}(x) \) of the function \( f(x) = \sqrt{3 + 3x} \), we can use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Here, the outer function is the square root function, and the inner function is \( 3 + 3x \).

The function given is \( f(x) = \sqrt{3 + 3x} \). This can be expressed as a composition of two functions: the outer function \( g(u) = \sqrt{u} \) and the inner function \( u(x) = 3 + 3x \).

To find the derivative \( f^{\prime}(x) \), we apply the chain rule. The chain rule states that if \( f(x) = g(u(x)) \), then \( f^{\prime}(x) = g^{\prime}(u(x)) \cdot u^{\prime}(x) \).

1. Differentiate the outer function: \( g(u) = \sqrt{u} \) gives \( g^{\prime}(u) = \frac{1}{2\sqrt{u}} \).
2. Differentiate the inner function: \( u(x) = 3 + 3x \) gives \( u^{\prime}(x) = 3 \).

Substitute back into the chain rule formula: \[ f^{\prime}(x) = g^{\prime}(u(x)) \cdot u^{\prime}(x) = \frac{1}{2\sqrt{3 + 3x}} \cdot 3 \]

Simplify the expression for the derivative: \[ f^{\prime}(x) = \frac{3}{2\sqrt{3 + 3x}} \]

Final Answer