What is the derivative of 5x(sin(x+cos(x))) ?

The derivative of 5x(sin(x+cos(x))) is 5(sin(x)+cos(x)+x(cos(x)-sin(x)))

What is the first derivative of 5x(sin(x+cos(x))) ?

The first derivative of 5x(sin(x+cos(x))) is 5(sin(x)+cos(x)+x(cos(x)-sin(x)))

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\frac{d}{d x} (5 x (sin (x) + cos (x)))

5 (sin (x) + cos (x) + x (cos (x) - sin (x)))

Solution steps

\frac{d}{d x} (5 x (sin (x) + cos (x)))

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

\frac{d x}{d x} = 1

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

= 5 (1 \cdot (sin (x) + cos (x)) + (cos (x) - sin (x)) x)

Simplify

= 5 (sin (x) + cos (x) + x (cos (x) - sin (x)))

7 x^{2} y + y^{^{'}} = 6 x^{2}

y^{^{'''}} - 5 y^{^{''}} - 6 y^{^{'}} = 0

\frac{d}{d t} (t)

x y^{^{'}} - 4 y = x

f (t) = cos (t)

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