Question

given that
$f(x)=x^{7}h(x)$
$h(-1)=5$
$h(-1)=8$
calculate $f(-1)$.
hint: use the product rule and the power rule.
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Explanation:

Step1: Apply the product rule

The product rule states that if \(f(x)=u(x)v(x)\), then \(f^{\prime}(x)=u^{\prime}(x)v(x)+u(x)v^{\prime}(x)\). Here \(u(x) = x^{7}\) and \(v(x)=h(x)\).
The power rule \((x^{n})^{\prime}=nx^{n - 1}\) gives \(u^{\prime}(x)=7x^{6}\). So \(f^{\prime}(x)=7x^{6}h(x)+x^{7}h^{\prime}(x)\).

Step2: Substitute \(x = - 1\)

Substitute \(x=-1\), \(h(-1) = 5\) and \(h^{\prime}(-1)=8\) into \(f^{\prime}(x)\).

$$
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Step3: Calculate the value

$$
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Answer:

\(27\)