Higher order derivatives

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Here we look at higher order derivatives.

We call the derivative of the derivative the second derivative, the derivative of the derivative of the derivative the third derivative, and so on. We have special notation for higher derivatives, check it out:

First derivative:: $\ddx f(x) = f'(x) = f^{(1)}(x)$ .
Second derivative:: $\dd [~^2]{x^2} f(x) = f''(x) = f^{(2)}(x)$ .
Third derivative:: $\dd [~^3]{x^3} f(x) = f'''(x) = f^{(3)}(x)$ .

Compute: $\dd [~^2]{x^2} e^{x^{2}}$

We need to first find $\ddx e^{x^{2}}$ $\ddx e^{x^{2}} = \answer [given]{2xe^{x^{2}}}\\$ Then $\dd [~^2]{x^2} e^{x^{2}} = \ddx 2xe^{x^{2}}$ $\begin{align*} \dd[~^2]{x^2} e^{x^{2}} &= \ddx 2xe^{x^{2}}\\ &= 2x \answer[given]{ 2xe^{x^{2}} }+ \answer[given]{2} 2xe^{x^{2}} \end{align*}$

Compute: $\dd [~^2]{x^2} \ln (x^{2}+2x+10)$

We need to first find $\ddx \ln (x^{2}+3x+10)$ $\ddx \ln (x^{2}+3x+10) = \answer [given]{\frac {2x+3}{x^{2}+3x+10}}\\$ Then $\dd [~^2]{x^2} \ln (x^{2}+3x+10) = \ddx \frac {2x+3}{x^{2}+3x+10}$ $\begin{align*} \dd[~^2]{x^2} \ln (x^{2}+3x+10) &= \ddx \frac{2x+3}{x^{2}+3x+10}\\ &= \frac{ \answer[given]{2}(x^{2}+3x+10)+ (2x+3)\answer[given]{2x+3} }{ \answer[given]{(x^{2}+3x+10)^{2} }} \end{align*}$