let's consider F(x)= \[a^{mx}\]
on differentiating both sides w.r.t x we get \[\frac{\mathrm{d}{ } }{\mathrm{d} x} a^{mx}\]= \[ a^{mx}m log_{e}(a)\]
On differentiating once again w.r.t x we get Dif\[\frac{\mathrm{d^2}{ a^{mx}} }{\mathrm{d} x^2} \]=\[ a^{mx}m^2 (log_{e}(a))^2\]
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Finally \[\frac{\mathrm{d^n}{ a^{mx}} }{\mathrm{d} x^n} \]=\[a^{mx}m^n (log_{e}(a))^n\]
Problem. Find the value of 4th derivative of \[3^{2x}\] at x=0.5 ?
Solution: \[\frac{\mathrm{d^4}{( 3^{2x})} }{\mathrm{d} x^4} \]=\[(3^{0.5*2})(2^4) (log_{e}(3))^4\] =69.92
