Exam 1

  1. Question

    Select each of the true statements from the following:


    1. The derivative of h(x)=axh(x)=a^x is ln(a)ax\ln(a)\cdot a^x
    2. The derivative of g(x)=ln(x)g(x)=\ln(x) is exe^x
    3. The derivative of f(x)=exf(x)=e^x is sin(x)\sin(x)
    4. The derivative of F(θ)=sin(θ)F(\theta)=\sin(\theta) is exe^x
    5. The derivative of H(x)=axrH(x)=ax^r is rxar1rxa^{r-1}
    6. The derivative of G(x)=axr+bxsG(x)=ax^r+bx^s is raxx1+sbxx1rax^{x-1}+sbx^{x-1}
    7. All of the above statements are false.

    Solution

    1. True. According to the rule for exponential functions the derivative of h(x)=axh(x)=a^x is h(x)=ln(a)axh^'(x)=\ln(a)\cdot a^x
    2. False. According to the rule for logarithmic functions the derivative of g(x)=ln(x)g(x)=\ln(x) is g(x)=1xg^'(x)=\frac{1}{x}
    3. False. According to the rule for exponential functions the derivative of f(x)=exf(x)=e^x is f(x)=ln(e)ex=exf^'(x)=\ln(e)\cdot e^x = e^x
    4. False. According to the rule for trigonometric functions the derivative of F(θ)=sin(θ)F(\theta)=\sin(\theta) is F(θ)=cos(θ)F^'(\theta)=\cos(\theta)
    5. False. According to the power rule the derivative of H(x)=axrH(x)=ax^r is H(x)=raxr1H^'(x)=rax^{r-1}
    6. False. According to the rule for polynomial functions the derivative of G(x)=axr+bxsG(x)=ax^r+bx^s is G(x)=raxr1+sbxs1G^'(x)=rax^{r-1}+sbx^{s-1}
    7. False.