What is the -derivative of tan(3x) ?

The -derivative of tan(3x) is -x derivative of tan(3x)+3xtan^2(3x)+3x-tan(3x)+C

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Find the Derivative - d/dx tan(3x)

\int - \frac{d}{d x} (tan (3 x)) d x

- x \frac{d}{d x} (tan (3 x)) + 3 x tan^{2} (3 x) + 3 x - tan (3 x) + C

Solution steps

\int - \frac{d}{d x} (tan (3 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} (tan (3 x)) d x

Apply Integration By Parts

: \frac{d}{d x} (tan (3 x)) x - \int 18 sec^{2} (3 x) tan (3 x) x d x

= - (\frac{d}{d x} (tan (3 x)) x - \int 18 sec^{2} (3 x) tan (3 x) x d x)

\int 18 sec^{2} (3 x) tan (3 x) x d x = 18 (\frac{1}{6} x tan^{2} (3 x) - \frac{1}{6} (- x + \frac{1}{3} tan (3 x)))

= - (\frac{d}{d x} (tan (3 x)) x - 18 (\frac{1}{6} x tan^{2} (3 x) - \frac{1}{6} (- x + \frac{1}{3} tan (3 x))))

Simplify

- (\frac{d}{d x} (tan (3 x)) x - 18 (\frac{1}{6} x tan^{2} (3 x) - \frac{1}{6} (- x + \frac{1}{3} tan (3 x)))) : - x \frac{d}{d x} (tan (3 x)) + 3 x tan^{2} (3 x) + 3 x - tan (3 x)

= - x \frac{d}{d x} (tan (3 x)) + 3 x tan^{2} (3 x) + 3 x - tan (3 x)

Add a constant to the solution

= - x \frac{d}{d x} (tan (3 x)) + 3 x tan^{2} (3 x) + 3 x - tan (3 x) + C

- \frac{d}{d x} f (x) = e^{4 x}

- \frac{d}{d x} y = 6 x

- \frac{d}{d x} (7 x)

\frac{d y}{d x} y = tan (x)

- \frac{d}{d x} (tan^{3} (x))

What is the -derivative of tan(3x) ?
The -derivative of tan(3x) is -x derivative of tan(3x)+3xtan^2(3x)+3x-tan(3x)+C