What is the -d/(dt)(e^t) ?

The -d/(dt)(e^t) is -t d/(dt)(e^t)+e^tt-e^t+C

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Find the Derivative - d/dt e^t

\int - \frac{d}{d t} (e^{t}) d t

- t \frac{d}{d t} (e^{t}) + e^{t} t - e^{t} + C

Solution steps

\int - \frac{d}{d t} (e^{t}) d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{d}{d t} (e^{t}) d t

Apply Integration By Parts

: \frac{d}{d t} (e^{t}) t - \int e^{t} t d t

= - (\frac{d}{d t} (e^{t}) t - \int e^{t} t d t)

\int e^{t} t d t = e^{t} t - e^{t}

= - (\frac{d}{d t} (e^{t}) t - (e^{t} t - e^{t}))

Simplify

- (\frac{d}{d t} (e^{t}) t - (e^{t} t - e^{t})) : - t \frac{d}{d t} (e^{t}) + e^{t} t - e^{t}

= - t \frac{d}{d t} (e^{t}) + e^{t} t - e^{t}

Add a constant to the solution

= - t \frac{d}{d t} (e^{t}) + e^{t} t - e^{t} + C

\int \frac{1}{( y ^{2} )} d y

- \frac{d}{d x} (\frac{( e ^{x} )}{x})

x \to \infty lim (\frac{x , x , ( x + x ^{2} )}{( 2 x - x ^{2} )})

- \frac{d}{d x} y = \frac{7}{x}

- \frac{d}{d x} (\frac{8}{x})