Question

find the value of (f ∘ g) at the given value. f(u)=u^5 - 3, u = g(x)=\sqrt{x}, x = 1. (f ∘ g)(1)= (type an integer or a simplified fraction.)

Explanation:

Step1: Find $f'(u)$

Differentiate $f(u)=u^{5}-3$ with respect to $u$. Using the power - rule $\frac{d}{du}(u^{n})=nu^{n - 1}$, we get $f'(u)=5u^{4}$.

Step2: Find $g'(x)$

Differentiate $g(x)=\sqrt{x}=x^{\frac{1}{2}}$ with respect to $x$. Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we get $g'(x)=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}$.

Step3: Use the chain - rule

The chain - rule states that $(f\circ g)'(x)=f'(g(x))\cdot g'(x)$.
First, find $g(1)$: $g(1)=\sqrt{1}=1$.
Then, find $f'(g(1))$: Since $g(1) = 1$, then $f'(g(1))=f'(1)$. Substitute $u = 1$ into $f'(u)$, we get $f'(1)=5\times1^{4}=5$.
Next, find $g'(1)$: Substitute $x = 1$ into $g'(x)$, we get $g'(1)=\frac{1}{2\sqrt{1}}=\frac{1}{2}$.
Finally, $(f\circ g)'(1)=f'(g(1))\cdot g'(1)$.
$(f\circ g)'(1)=5\times\frac{1}{2}=\frac{5}{2}$.

Answer:

$\frac{5}{2}$