Sovi.AI - AI Math Tutor

Scan to solve math questions
ENFRDEESJAZH-CNZH-TW

QUESTION IMAGE

68 - 72. higher - order derivatives find $f(x), f(x),$ and $f(x)$ for t…

Question

68 - 72. higher - order derivatives find $f(x), f(x),$ and $f(x)$ for the following functions.

  1. $f(x)=3x^{3}+5x^{2}+6x$
  2. $f(x)=5x^{4}+10x^{3}+3x + 6$
  3. $f(x)=3x^{2}+5e^{x}$
  4. $f(x)=\frac{x^{2}-7x - 8}{x + 1}$
  5. $f(x)=10e^{x}$

Explanation:

Response
  1. For \(f(x)=3x^{3}+5x^{2}+6x\):
  • Step - by - step to find \(f^{\prime}(x)\):
  • Step1: Apply the power rule \((x^n)^\prime = nx^{n - 1}\) to each term
  • The derivative of \(3x^{3}\) is \(3\times3x^{3 - 1}=9x^{2}\), the derivative of \(5x^{2}\) is \(5\times2x^{2 - 1}=10x\), and the derivative of \(6x\) is \(6\). So, \(f^{\prime}(x)=9x^{2}+10x + 6\).
  • Step - by - step to find \(f^{\prime\prime}(x)\):
  • Step1: Differentiate \(f^{\prime}(x)\) using the power rule
  • The derivative of \(9x^{2}\) is \(9\times2x^{2 - 1}=18x\), the derivative of \(10x\) is \(10\), and the derivative of the constant \(6\) is \(0\). So, \(f^{\prime\prime}(x)=18x + 10\).
  • Step - by - step to find \(f^{\prime\prime\prime}(x)\):
  • Step1: Differentiate \(f^{\prime\prime}(x)\) using the power rule
  • The derivative of \(18x\) is \(18\) and the derivative of the constant \(10\) is \(0\). So, \(f^{\prime\prime\prime}(x)=18\).
  1. For \(f(x)=5x^{4}+10x^{3}+3x + 6\):
  • Step - by - step to find \(f^{\prime}(x)\):
  • Step1: Apply the power rule \((x^n)^\prime=nx^{n - 1}\) to each term
  • The derivative of \(5x^{4}\) is \(5\times4x^{4 - 1}=20x^{3}\), the derivative of \(10x^{3}\) is \(10\times3x^{3 - 1}=30x^{2}\), the derivative of \(3x\) is \(3\), and the derivative of the constant \(6\) is \(0\). So, \(f^{\prime}(x)=20x^{3}+30x^{2}+3\).
  • Step - by - step to find \(f^{\prime\prime}(x)\):
  • Step1: Differentiate \(f^{\prime}(x)\) using the power rule
  • The derivative of \(20x^{3}\) is \(20\times3x^{3 - 1}=60x^{2}\), the derivative of \(30x^{2}\) is \(30\times2x^{2 - 1}=60x\), and the derivative of the constant \(3\) is \(0\). So, \(f^{\prime\prime}(x)=60x^{2}+60x\).
  • Step - by - step to find \(f^{\prime\prime\prime}(x)\):
  • Step1: Differentiate \(f^{\prime\prime}(x)\) using the power rule
  • The derivative of \(60x^{2}\) is \(60\times2x^{2 - 1}=120x\), and the derivative of \(60x\) is \(60\). So, \(f^{\prime\prime\prime}(x)=120x + 60\).
  1. For \(f(x)=3x^{2}+5e^{x}\):
  • Step - by - step to find \(f^{\prime}(x)\):
  • Step1: Apply the power rule to \(3x^{2}\) and the rule \((e^{x})^\prime = e^{x}\) to \(5e^{x}\)
  • The derivative of \(3x^{2}\) is \(3\times2x^{2 - 1}=6x\), and the derivative of \(5e^{x}\) is \(5e^{x}\). So, \(f^{\prime}(x)=6x + 5e^{x}\).
  • Step - by - step to find \(f^{\prime\prime}(x)\):
  • Step1: Differentiate \(f^{\prime}(x)\) using the power rule and \((e^{x})^\prime=e^{x}\)
  • The derivative of \(6x\) is \(6\), and the derivative of \(5e^{x}\) is \(5e^{x}\). So, \(f^{\prime\prime}(x)=6 + 5e^{x}\).
  • Step - by - step to find \(f^{\prime\prime\prime}(x)\):
  • Step1: Differentiate \(f^{\prime\prime}(x)\) using \((e^{x})^\prime=e^{x}\)
  • The derivative of the constant \(6\) is \(0\), and the derivative of \(5e^{x}\) is \(5e^{x}\). So, \(f^{\prime\prime\prime}(x)=5e^{x}\).
  1. For \(f(x)=\frac{x^{2}-7x - 8}{x + 1}\):
  • First, simplify \(f(x)\) by factoring the numerator: \(x^{2}-7x - 8=(x + 1)(x - 8)\), so \(f(x)=x - 8\) for \(x

eq - 1\).

  • Step - by - step to find \(f^{\prime}(x)\):
  • Step1: Apply the power rule \((x^n)^\prime=nx^{n - 1}\) (here \(n = 1\) for \(x\) and \(n = 0\) for the constant \(-8\))
  • The derivative of \(x\) is \(1\) and the derivative of the constant \(-8\) is \(0\). So, \(f^{\prime}(x)=1\).
  • Step - by - step to find \(f^{\prime\prime}(x)\):
  • Step1: Differentiate \(f^{\prime}(x)\)…

Answer:

  • For \(f(x)=3x^{3}+5x^{2}+6x\): \(f^{\prime}(x)=9x^{2}+10x + 6\), \(f^{\prime\prime}(x)=18x + 10\), \(f^{\prime\prime\prime}(x)=18\).
  • For \(f(x)=5x^{4}+10x^{3}+3x + 6\): \(f^{\prime}(x)=20x^{3}+30x^{2}+3\), \(f^{\prime\prime}(x)=60x^{2}+60x\), \(f^{\prime\prime\prime}(x)=120x + 60\).
  • For \(f(x)=3x^{2}+5e^{x}\): \(f^{\prime}(x)=6x + 5e^{x}\), \(f^{\prime\prime}(x)=6 + 5e^{x}\), \(f^{\prime\prime\prime}(x)=5e^{x}\).
  • For \(f(x)=\frac{x^{2}-7x - 8}{x + 1}\): \(f^{\prime}(x)=1\), \(f^{\prime\prime}(x)=0\), \(f^{\prime\prime\prime}(x)=0\).
  • For \(f(x)=10e^{x}\): \(f^{\prime}(x)=10e^{x}\), \(f^{\prime\prime}(x)=10e^{x}\), \(f^{\prime\prime\prime}(x)=10e^{x}\).