What is the -derivative of 6^x ?

The -derivative of 6^x is -x derivative of 6^x+ln(6)*6^xx-6^x+C

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Find the Derivative - d/dx 6^x

\int - \frac{d}{d x} (6^{x}) d x

- x \frac{d}{d x} (6^{x}) + ln (6) \cdot 6^{x} x - 6^{x} + C

Solution steps

\int - \frac{d}{d x} (6^{x}) d x

Take the constant out:

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

= - \int \frac{d}{d x} (6^{x}) d x

Apply Integration By Parts

: \frac{d}{d x} (6^{x}) x - \int ln^{2} (6) \cdot 6^{x} x d x

= - (\frac{d}{d x} (6^{x}) x - \int ln^{2} (6) \cdot 6^{x} x d x)

\int ln^{2} (6) \cdot 6^{x} x d x = ln (6) \cdot 6^{x} x - 6^{x}

= - (\frac{d}{d x} (6^{x}) x - (ln (6) \cdot 6^{x} x - 6^{x}))

Simplify

- (\frac{d}{d x} (6^{x}) x - (ln (6) \cdot 6^{x} x - 6^{x})) : - x \frac{d}{d x} (6^{x}) + ln (6) \cdot 6^{x} x - 6^{x}

= - x \frac{d}{d x} (6^{x}) + ln (6) \cdot 6^{x} x - 6^{x}

Add a constant to the solution

= - x \frac{d}{d x} (6^{x}) + ln (6) \cdot 6^{x} x - 6^{x} + C

sec (2 x)

- \frac{d}{d x} y = 3 x^{2}

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

\int (\frac{1}{( x - x ^{2} )}) d x

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

What is the -derivative of 6^x ?
The -derivative of 6^x is -x derivative of 6^x+ln(6)*6^xx-6^x+C