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    • 1.9.15: Derivatives
      https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/01%3A_Introduction_to_Physics_Measurements_and_Mathematics_Tools/1.09%3A_Math_Review_of_Other_Topics/1.9.15%3A_Derivatives
      We start by calculating \(\Delta f\): \[\begin{aligned} \Delta f &= f(x+\Delta x)-f(x)\\ &=(x+\Delta x)^2 - x^2\\ &=x^2+2x\Delta x+\Delta x^2 -x^2\\ &=2x\Delta x+\Delta x^2\end{aligned}\] We now calcu...We start by calculating \(\Delta f\): \[\begin{aligned} \Delta f &= f(x+\Delta x)-f(x)\\ &=(x+\Delta x)^2 - x^2\\ &=x^2+2x\Delta x+\Delta x^2 -x^2\\ &=2x\Delta x+\Delta x^2\end{aligned}\] We now calculate \(\frac{\Delta f}{\Delta x}\): \[\begin{aligned} \frac{\Delta f}{\Delta x}&=\frac{2x\Delta x+\Delta x^2}{\Delta x}\\ &=2x+\Delta x\end{aligned}\] and take the limit \(\Delta x\to 0\): \[\begin{aligned} \frac{df}{dx}&=\lim_{\Delta x\to 0 }\frac{\Delta f}{\Delta x}\\ &=\lim_{\Delta x\to 0 }(2x+\…
    • 26.2: Derivatives
      https://phys.libretexts.org/Courses/Berea_College/Introductory_Physics%3A_Berea_College/26%3A_Calculus/26.02%3A_Derivatives
      We start by calculating \(\Delta f\): \[\begin{aligned} \Delta f &= f(x+\Delta x)-f(x)\\ &=(x+\Delta x)^2 - x^2\\ &=x^2+2x\Delta x+\Delta x^2 -x^2\\ &=2x\Delta x+\Delta x^2\end{aligned}\] We now calcu...We start by calculating \(\Delta f\): \[\begin{aligned} \Delta f &= f(x+\Delta x)-f(x)\\ &=(x+\Delta x)^2 - x^2\\ &=x^2+2x\Delta x+\Delta x^2 -x^2\\ &=2x\Delta x+\Delta x^2\end{aligned}\] We now calculate \(\frac{\Delta f}{\Delta x}\): \[\begin{aligned} \frac{\Delta f}{\Delta x}&=\frac{2x\Delta x+\Delta x^2}{\Delta x}\\ &=2x+\Delta x\end{aligned}\] and take the limit \(\Delta x\to 0\): \[\begin{aligned} \frac{df}{dx}&=\lim_{\Delta x\to 0 }\frac{\Delta f}{\Delta x}\\ &=\lim_{\Delta x\to 0 }(2x+\…
    • 1.9.15: Derivatives
      https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/01%3A_Introduction_to_Physics_Measurements_and_Mathematics_Tools/1.09%3A_Math_Review_of_Other_Topics/1.9.15%3A_Derivatives
      We start by calculating \(\Delta f\): \[\begin{aligned} \Delta f &= f(x+\Delta x)-f(x)\\ &=(x+\Delta x)^2 - x^2\\ &=x^2+2x\Delta x+\Delta x^2 -x^2\\ &=2x\Delta x+\Delta x^2\end{aligned}\] We now calcu...We start by calculating \(\Delta f\): \[\begin{aligned} \Delta f &= f(x+\Delta x)-f(x)\\ &=(x+\Delta x)^2 - x^2\\ &=x^2+2x\Delta x+\Delta x^2 -x^2\\ &=2x\Delta x+\Delta x^2\end{aligned}\] We now calculate \(\frac{\Delta f}{\Delta x}\): \[\begin{aligned} \frac{\Delta f}{\Delta x}&=\frac{2x\Delta x+\Delta x^2}{\Delta x}\\ &=2x+\Delta x\end{aligned}\] and take the limit \(\Delta x\to 0\): \[\begin{aligned} \frac{df}{dx}&=\lim_{\Delta x\to 0 }\frac{\Delta f}{\Delta x}\\ &=\lim_{\Delta x\to 0 }(2x+\…

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