What is the -derivative of 4sin(x) ?

The -derivative of 4sin(x) is -x derivative of 4sin(x)+4xcos(x)-4sin(x)+C

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Find the Derivative - d/dx 4sin(x)

\int - \frac{d}{d x} (4 sin (x)) d x

- x \frac{d}{d x} (4 sin (x)) + 4 x cos (x) - 4 sin (x) + C

Solution steps

\int - \frac{d}{d x} (4 sin (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} (4 sin (x)) d x

Apply Integration By Parts

: x \frac{d}{d x} (4 sin (x)) - \int - 4 x sin (x) d x

= - (x \frac{d}{d x} (4 sin (x)) - \int - 4 x sin (x) d x)

\int - 4 x sin (x) d x = - 4 (- x cos (x) + sin (x))

= - (x \frac{d}{d x} (4 sin (x)) - (- 4 (- x cos (x) + sin (x))))

Simplify

- (x \frac{d}{d x} (4 sin (x)) - (- 4 (- x cos (x) + sin (x)))) : - x \frac{d}{d x} (4 sin (x)) + 4 x cos (x) - 4 sin (x)

= - x \frac{d}{d x} (4 sin (x)) + 4 x cos (x) - 4 sin (x)

Add a constant to the solution

= - x \frac{d}{d x} (4 sin (x)) + 4 x cos (x) - 4 sin (x) + C

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

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

- \frac{d}{d x} (4)

- \frac{d}{d x} (csc (2 x))

- \frac{d}{d x} y = \frac{1}{( x - 1 )}

What is the -derivative of 4sin(x) ?
The -derivative of 4sin(x) is -x derivative of 4sin(x)+4xcos(x)-4sin(x)+C