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total_derivative [2015/03/28 16:28] nikolaj |
total_derivative [2015/03/28 16:28] nikolaj |
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| === Example === | === Example === | ||
| - | The total derivative of the function | + | The total derivative of the (not explicitly time dependent) function |
| $f:{\mathbb R}^3\to{\mathbb R}, \ \ \ f(x,y,t):=x^2\cos(y)$ | $f:{\mathbb R}^3\to{\mathbb R}, \ \ \ f(x,y,t):=x^2\cos(y)$ | ||
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| $\left(\dfrac{{\mathrm d}}{{\mathrm d}t}f\right)(R,t)=2\,R_x(t)\cos(R_y(t))\cdot R_x'(t)+R_x(t)^2\sin(R_y(t))\cdot R_y'(t)$ | $\left(\dfrac{{\mathrm d}}{{\mathrm d}t}f\right)(R,t)=2\,R_x(t)\cos(R_y(t))\cdot R_x'(t)+R_x(t)^2\sin(R_y(t))\cdot R_y'(t)$ | ||
| - | $=2\,7^2\,t\cos(-3t^5)-3\,5\,7^2)\,t^{2+4}\sin(-3t^5)$ | + | $=2\cdot 7^2\,t\cos(-3t^5)-3\cdot 5\cdot 7^2\,t^{2+4}\sin(-3t^5)$ |
| === Reference === | === Reference === | ||
