What is the derivative of-5sin(x(sqrt(x)+3sin(x))) ?

The derivative of-5sin(x(sqrt(x)+3sin(x))) is -5(sqrt(x)cos(x)+3sin(2x)+(sin(x))/(2sqrt(x)))

What is the first derivative of-5sin(x(sqrt(x)+3sin(x))) ?

The first derivative of-5sin(x(sqrt(x)+3sin(x))) is -5(sqrt(x)cos(x)+3sin(2x)+(sin(x))/(2sqrt(x)))

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\frac{d}{d x} (- 5 sin (x) (x + 3 sin (x)))

- 5 (x cos (x) + 3 sin (2 x) + \frac{sin ( x )}{2 x})

Solution steps

\frac{d}{d x} (- 5 sin (x) (x + 3 sin (x)))

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= - 5 \frac{d}{d x} (sin (x) (x + 3 sin (x)))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = sin (x), g = x + 3 sin (x)

= - 5 (\frac{d}{d x} (sin (x)) (x + 3 sin (x)) + \frac{d}{d x} (x + 3 sin (x)) sin (x))

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

\frac{d}{d x} (x + 3 sin (x)) = \frac{1}{2 x} + 3 cos (x)

= - 5 (cos (x) (x + 3 sin (x)) + (\frac{1}{2 x} + 3 cos (x)) sin (x))

Simplify

- 5 (cos (x) (x + 3 sin (x)) + (\frac{1}{2 x} + 3 cos (x)) sin (x)) : - 5 (x cos (x) + 3 sin (2 x) + \frac{sin ( x )}{2 x})

= - 5 (x cos (x) + 3 sin (2 x) + \frac{sin ( x )}{2 x})

(x^{9} - 2)^{3}

x \to 1 lim (2 - x)

\frac{\partial}{\partial x} (\frac{2 x y}{( x ^{2} - y ^{2} ) ^{2}})

y^{^{'}} - \frac{3}{x} y = 4 x^{3} cos (2 x)

f (x) = x^{\frac{1}{3}}

What is the derivative of-5sin(x(sqrt(x)+3sin(x))) ?
The derivative of-5sin(x(sqrt(x)+3sin(x))) is -5(sqrt(x)cos(x)+3sin(2x)+(sin(x))/(2sqrt(x)))
What is the first derivative of-5sin(x(sqrt(x)+3sin(x))) ?
The first derivative of-5sin(x(sqrt(x)+3sin(x))) is -5(sqrt(x)cos(x)+3sin(2x)+(sin(x))/(2sqrt(x)))