`f(x) = ln(sinhx)` Find the derivative of the function

Sunday, April 7, 2013

$\displaystyle{f{{\left({x}\right)}}}={\ln{{\left({\sinh{{x}}}\right)}}}$ Find the derivative of the function

$\displaystyle{f{{\left({x}\right)}}}={\ln{{\left({\sinh{{\left({x}\right)}}}\right)}}}$

Take note that the derivative formula of natural logarithm is

$\displaystyle\frac{{d}}{{\left.{d}{x}}}{\left[{\ln{{\left({u}\right)}}}\right]}=\frac{{1}}{{u}}\cdot\frac{{{d}{u}}}{{\left.{d}{x}}}$

Applying this formula, the derivative of the function will be

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{d}}{{\left.{d}{x}}}{\left[{\ln{{\left({\sinh{{\left({x}\right)}}}\right)}}}\right]}$

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{1}}{{{\sinh{{\left({x}\right)}}}}}\cdot\frac{{d}}{{\left.{d}{x}}}{\left[{\sinh{{\left({x}\right)}}}\right]}$

To take the derivative of hyperbolic sine, apply the formula

$\displaystyle\frac{{d}}{{\left.{d}{x}}}{\left[{\sinh{{\left({u}\right)}}}\right]}={\cosh{{\left({u}\right)}}}\cdot\frac{{{d}{u}}}{{\left.{d}{x}}}$

So f'(x) will become

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{1}}{{{\sinh{{\left({x}\right)}}}}}\cdot{\cosh{{\left({x}\right)}}}\cdot\frac{{d}}{{\left.{d}{x}}}{\left({x}\right)}$

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{1}}{{{\sinh{{\left({x}\right)}}}}}\cdot{\cosh{{\left({x}\right)}}}\cdot{1}$

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{\cosh{{\left({x}\right)}}}}{{\sinh{{\left({x}\right)}}}}$

Since the ratio of hyperbolic cosine to hyperbolic sine is equal to hyperbolic cotangent, the f'(x) will simplify to

$\displaystyle{f{'}}{\left({x}\right)}={\coth{{\left({x}\right)}}}$

Therefore, the derivative of the given function is $\displaystyle{f{'}}{\left({x}\right)}={\coth{{\left({x}\right)}}}$ .

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Sunday, April 7, 2013

$\displaystyle{f{{\left({x}\right)}}}={\ln{{\left({\sinh{{x}}}\right)}}}$ Find the derivative of the function

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Sunday, April 7, 2013

Find the derivative of the function

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Post a Comment

Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle{f{{\left({x}\right)}}}={\ln{{\left({\sinh{{x}}}\right)}}}$ Find the derivative of the function

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?