Question

find f(-4) if f(x) = \frac{x^{3}}{7}-5x.
f(-4)= (simplify your answer. type an integer or a fraction.)

Explanation:

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

Use the power - rule \((x^n)^\prime=nx^{n - 1}\). For \(y=\frac{x^{3}}{7}-5x\), the derivative \(f^\prime(x)=\frac{3x^{2}}{7}-5\).

Step2: Substitute \(x = - 4\) into \(f^\prime(x)\)

\(f^\prime(-4)=\frac{3\times(-4)^{2}}{7}-5\). First, calculate \((-4)^{2}=16\). Then \(3\times(-4)^{2}=3\times16 = 48\). So \(f^\prime(-4)=\frac{48}{7}-5\).

Step3: Rewrite 5 with a denominator of 7

\(5=\frac{35}{7}\), then \(f^\prime(-4)=\frac{48}{7}-\frac{35}{7}=\frac{48 - 35}{7}=\frac{13}{7}\).

Answer:

\(\frac{13}{7}\)