Question:

Let $f(x)$ and $g(u)$ be differentiable functions. Given that $f(x) = 4x^3 - 2x^2 + 5x - 7$ and $g(u) = \sqrt{3u + 2}$, find the derivative of $h(x) = g[f(x)]$ using the chain rule.

Answer:

To find the derivative of $h(x)$, we can use the chain rule. The chain rule states that if we have a composite function $h(x) = g[f(x)]$, where $f(x)$ represents the inner function and $g(u)$ represents the outer function, then the derivative is given by:

First, we need to find $\frac{dg}{du}$ and $\frac{df}{dx}$.

Given $g(u) = \sqrt{3u + 2}$, let's find $\frac{dg}{du}$:

Using the power rule and the chain rule, we have:

Simplifying the derivative of $3u+2$, we get:

Now, let's find $\frac{df}{dx}$:

Given $f(x) = 4x^3 - 2x^2 + 5x - 7$, we simply need to differentiate $f(x)$ using the power rule:

Now that we have both $\frac{dg}{du}$ and $\frac{df}{dx}$, we can find $\frac{dh}{dx}$:

Plugging in the values we found earlier:

However, we need to substitute $u$ back in terms of $x$ using the inner function $f(x)$:

Substituting this back into the equation:

Simplifying further gives us the final derivative expression:

Therefore, the derivative of $h(x) = g[f(x)]$ with respect to $x$ is $\frac{3(12x^2 - 4x + 5)}{2\sqrt{12x^3 - 6x^2 + 15x -16}}$.