Question

find the value of (f ◦ g)’ at the given value.
f(u)=u^{9}-5, u = g(x)=\sqrt{x}, x = 1
(f ◦ g)’(1)=
(type an integer or a simplified fraction.)

Explanation:

Step1: Find the derivative of \(f(u)\)

Using the power rule \((x^n)^\prime=nx^{n - 1}\), for \(f(u)=u^{9}-5\), \(f^\prime(u)=9u^{8}\)

Step2: Find the derivative of \(g(x)\)

For \(g(x)=\sqrt{x}=x^{\frac{1}{2}}\), using the power rule \((x^n)^\prime=nx^{n - 1}\), \(g^\prime(x)=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}\)

Step3: Apply the chain - rule \((f\circ g)^\prime(x)=f^\prime(g(x))\cdot g^\prime(x)\)

Substitute \(u = g(x)\) into \(f^\prime(u)\), we get \(f^\prime(g(x))=9(g(x))^{8}=9x^{4}\) (since \(g(x)=\sqrt{x}=x^{\frac{1}{2}}\), then \((g(x))^{8}=x^{4}\))
Then \((f\circ g)^\prime(x)=9x^{4}\cdot\frac{1}{2\sqrt{x}}=\frac{9}{2}x^{\frac{7}{2}}\)

Step4: Evaluate \((f\circ g)^\prime(x)\) at \(x = 1\)

Substitute \(x = 1\) into \((f\circ g)^\prime(x)\), \((f\circ g)^\prime(1)=\frac{9}{2}(1)^{\frac{7}{2}}\)

Answer:

\(\frac{9}{2}\)