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