Find the value of $(f og)'$ at $x = 4$ if $f(u) = u^3 + 1$ and $u = g(x) = \sqrt{x}$.

3

The correct answer is Option (1) → 3 ##

We have,

$g(x) = \sqrt{x} = x^{\frac{1}{2}}$

Differentiate $g(x)$:

$g'(x) = \frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}$

$f(u) = u^3 + 1$

Differentiating $f(u)$:

$f'(u) = 3u^2$

The derivative of the composite function $(f og)(x)$ is given by the chain rule:

$(f og)'(x) = f'(g(x)) \cdot g'(x)$

Now, need to evaluate $(f og)'(4)$:

Find $g(4)$:

$g(4) = \sqrt{4} = 2$

Find $f'(g(4)) = f'(2)$:

$f'(2) = 3(2^2) = 3 \times 4 = 12$

Find $g'(4)$:

$g'(4) = \frac{1}{2\sqrt{4}} = \frac{1}{2 \times 2} = \frac{1}{4}$

$(f o g)'(4) = f'(g(4)) \cdot g'(4) = 12 \cdot \frac{1}{4} = 3$