+65 Can someone help me use the definition of a derivative using the f(x) listed below.
\(f(x)=\frac{1}{\sqrt{x^2+5}}\)
I got my \(f(x+h)=\frac{1}{\sqrt{(x+h)^2+5}}\)
then subtracted my f(x+h) by my f(x) all over h and got confused at a few part and got the wrong answer. I know for a fact that my f(x+h) is correct tho.
+26367 Can someone help me use the definition of a derivative using the f(x) listed below.
\(f(x)=\frac{1}{\sqrt{x^2+5}}\)
\(\begin{array}{|rcll|} \hline f'(x) &=& \lim \limits_{h\to 0} \frac{ f(x+h) - f(x) } {h} \\\\ f'(x) &=& \lim \limits_{h\to 0} \frac{ \left( \frac{1}{ \sqrt{(x+h)^2+5} } - \frac{1}{ \sqrt{x^2+5}} \right) \cdot \left( \frac{1}{ \sqrt{(x+h)^2+5} } + \frac{1}{ \sqrt{x^2+5}} \right) } {h \cdot \left( \frac{1}{ \sqrt{(x+h)^2+5} } + \frac{1}{ \sqrt{x^2+5}} \right) } \quad | \quad (a-b)(a+b) = a^2-b^2\\ f'(x) &=& \lim \limits_{h\to 0} \frac{ \left( \frac{1}{ (x+h)^2+5 } - \frac{1}{ x^2+5 } \right) } {h \cdot \left( \frac{1}{ \sqrt{(x+h)^2+5} } + \frac{1}{ \sqrt{x^2+5}} \right) } \\ f'(x) &=& \lim \limits_{h\to 0} \frac{ \frac{(x^2+5)-[(x+h)^2+5] }{ [(x+h)^2+5]\cdot(x^2+5) } } {h \cdot \left( \frac{1}{ \sqrt{(x+h)^2+5} } + \frac{1}{ \sqrt{x^2+5}} \right) } \\ f'(x) &=& \lim \limits_{h\to 0} \frac{ \frac{(x^2+5)-(x^2+2xh+h^2+5) }{ [(x+h)^2+5]\cdot(x^2+5) } } {h \cdot \left( \frac{1}{ \sqrt{(x+h)^2+5} } + \frac{1}{ \sqrt{x^2+5}} \right) } \\ f'(x) &=& \lim \limits_{h\to 0} \frac{ \frac{(x^2+5)-x^2-2xh-h^2-5) }{ [(x+h)^2+5]\cdot(x^2+5) } } {h \cdot \left( \frac{1}{ \sqrt{(x+h)^2+5} } + \frac{1}{ \sqrt{x^2+5}} \right) } \\ f'(x) &=& \lim \limits_{h\to 0} \frac{ \frac{x^2+5-x^2-2xh-h^2-5 }{ [(x+h)^2+5]\cdot(x^2+5) } } {h \cdot \left( \frac{1}{ \sqrt{(x+h)^2+5} } + \frac{1}{ \sqrt{x^2+5}} \right) } \\ f'(x) &=& \lim \limits_{h\to 0} \frac{ \frac{ -2xh-h^2 }{ [(x+h)^2+5]\cdot(x^2+5) } } {h \cdot \left( \frac{1}{ \sqrt{(x+h)^2+5} } + \frac{1}{ \sqrt{x^2+5}} \right) } \\ f'(x) &=& \lim \limits_{h\to 0} \frac{ -2xh-h^2 } {h \cdot \left( \frac{1}{ \sqrt{(x+h)^2+5} } + \frac{1}{ \sqrt{x^2+5}} \right) \cdot [(x+h)^2+5]\cdot(x^2+5) } \\ f'(x) &=& \lim \limits_{h\to 0} \frac{ h\cdot(-2x-h) } {h \cdot \left( \frac{1}{ \sqrt{(x+h)^2+5} } + \frac{1}{ \sqrt{x^2+5}} \right) \cdot [(x+h)^2+5]\cdot(x^2+5) } \\ f'(x) &=& \lim \limits_{h\to 0} \frac{ -2x-h } { \left( \frac{1}{ \sqrt{(x+h)^2+5} } + \frac{1}{ \sqrt{x^2+5}} \right) \cdot [(x+h)^2+5]\cdot(x^2+5) } \\ f'(x) &=& \frac{ -2x-0 } { \left( \frac{1}{ \sqrt{(x+0)^2+5} } + \frac{1}{ \sqrt{x^2+5}} \right) \cdot [(x+0)^2+5]\cdot(x^2+5) } \\ f'(x) &=& \frac{ -2x } { \left( \frac{1}{ \sqrt{x^2+5} } + \frac{1}{ \sqrt{x^2+5}} \right) \cdot (x^2+5)\cdot(x^2+5) } \\ f'(x) &=& \frac{ -2x } { 2\cdot \frac{1}{ \sqrt{x^2+5} } \cdot (x^2+5)^2 } \\ f'(x) &=& \frac{ - x } { \frac{1}{ \sqrt{x^2+5} } \cdot (x^2+5)^2 } \\ f'(x) &=& \frac{ -x } { \frac{(x^2+5)^2}{ \sqrt{x^2+5} } } \\ f'(x) &=& \frac{ -x } { \frac{ \sqrt{(x^2+5)^4}} { \sqrt{x^2+5} } } \\ f'(x) &=& \frac{ -x } { \sqrt{ \frac{ (x^2+5)^4 } { x^2+5} } } \\ f'(x) &=& \frac{ -x } { \sqrt{ (x^2+5)^3 } } \\ \hline \end{array} \)
