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Mathematics LibreTexts

3.9 E: Derivatives Ln, etc. Exercises

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    10870
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    3.9: Derivatives of Ln, General Exponent & Log Functions; and Logarithmic Differentiation

    Exercise:

     

     

    For the following exercises, find \(f′(x)\) for each function.

    333) \(f(x)=e^{x^3lnx}\)

    Answer:

    \(e^{x^3}lnx(3x^2lnx+x^2)\)

    336) \(f(x)=\frac{10^x}{ln10}\)

    337) \(f(x)=2^{4x}+4x^2\)

    Answer:

    \(2^{4x+2}⋅ln2+8x\)

    338) \(f(x)=3^{sin3x}\)

    339) \(f(x)=x^π⋅π^x\)

    Answer:

    \(πx^{π−1}⋅π^x+x^π⋅π^xlnπ\)

    340) \(f(x)=ln(4x^3+x)\)

    341) \(f(x)=ln\sqrt{5x−7}\)

    Answer:

    \(\frac{5}{2(5x−7)}\)

    342) \(f(x)=x^2ln9x\)

    343) \(f(x)=log(secx)\)

    Answer:

    \(\frac{tanx}{ln10}\)

    344) \(f(x)=log_7(6x^4+3)^5\)

    345) \(f(x)=2^x⋅log_37^{x^2−4}\)

    Answer:

    \(2^x⋅ln2⋅log_37^{x^2−4}+2^x⋅\frac{2xln7}{ln3}\)

     

    For the following exercises, use logarithmic differentiation to find \(\frac{dy}{dx}\).

    346) \(y=x^{\sqrt{x}}\)

    347) \(y=(sin2x)^{4x}\)

    Answer:

    \((sin2x)^{4x}[4⋅ln(sin2x)+8x⋅cot2x]\)

    348) \(y=(lnx)^{lnx}\)

    349) \(y=x^{log_2x}\)

    Answer:

    \(x^{log_2x}⋅\frac{2lnx}{xln2}\)

    350) \(y=(x^2−1)^{lnx}\)

    351) \(y=x^{cotx}\)

    Answer:

    \(x^{cotx}⋅[−csc^2x⋅lnx+\frac{cotx}{x}]\)

    352) \(y=\frac{x+11}{\sqrt[3]{x^2−4}}\)

    353) \(y=x^{−1/2}(x^2+3)^{2/3}(3x−4)^4\)

    Answer:

    \(x^{−1/2}(x2+3)^{2/3}(3x−4)^4⋅[\frac{−1}{2x}+\frac{4x}{3(x^2+3)}+\frac{12}{3x−4}]\)

     

    354) [T] Find an equation of the tangent line to the graph of \(f(x)=4xe^{(x^2−1)}\) at the point where

    \(x=−1.\) Graph both the function and the tangent line.

    355) [T] Find the equation of the line that is normal to the graph of \(f(x)=x⋅5^x\) at the point where \(x=1\). Graph both the function and the normal line.

    Answer:

    \(y=\frac{−1}{5+5ln5}x+(5+\frac{1}{5+5ln5})\)

    356) [T] Find the equation of the tangent line to the graph of \(x^3−xlny+y^3=2x+5\) at the point where \(x=2\). (Hint: Use implicit differentiation to find \(\frac{dy}{dx}\).) Graph both the curve and the tangent line.

     

    J357) 

    use the graph of \(y=f(x)\) (shown below) to

    a. sketch the graph of \(y=f^{−1}(x)\), and

    b. use part a. to estimate \((f^{−1})′(1)\).

     

     

    Answer:

    a.

      

    b. \((f^{−1})′(1)~2\)

     

    For the next set of exercises, find \(\frac{dy}{dx}\).  [Hint: first take the ln of both sides.]

    J358)  \(y=\frac{(2x^3−15x)\sqrt{6x^{4}+7}}{3x^2−x+3}\)

     

    J359)  \(y={30x^4}\sqrt{17x+2}{(sin(x))}\)

     

    J360)  \(y={e^{5x}}{(3x-1)^\frac{2}{3}}{(8^{3x})}\)