Question

if ( f(x)=\frac{7 e^{x}}{11}-\frac{9 x^{5}}{16}+2 ), find ( f^{prime}(x) ).

Explanation:

Step1: Differentiate each term separately

Use the rules: \((e^x)^\prime = e^x\), \((x^n)^\prime=nx^{n - 1}\), and \((C)^\prime = 0\) (where \(C\) is a constant).
For the first term \(\frac{7e^x}{11}\), its derivative is \(\frac{7}{11}(e^x)^\prime\).
For the second term \(-\frac{9x^5}{16}\), its derivative is \(-\frac{9}{16}(x^5)^\prime\).
For the third term \(2\), its derivative is \(0\).

Step2: Apply the power rule and the derivative of \(e^x\)

• For \(\frac{7}{11}(e^x)^\prime\): Since \((e^x)^\prime = e^x\), this term becomes \(\frac{7e^x}{11}\).
• For \(-\frac{9}{16}(x^5)^\prime\): Using \((x^n)^\prime=nx^{n - 1}\), we have \(-\frac{9}{16}\times5x^{5 - 1}=-\frac{45x^4}{16}\).

Answer:

\(f^\prime(x)=\frac{7e^x}{11}-\frac{45x^4}{16}\)