What is the derivative of 2x(4-x^2^{1/2}) ?

The derivative of 2x(4-x^2^{1/2}) is (2(-2x^2+4))/((4-x^2)^{1/2)}

What is the first derivative of 2x(4-x^2^{1/2}) ?

The first derivative of 2x(4-x^2^{1/2}) is (2(-2x^2+4))/((4-x^2)^{1/2)}

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\frac{d}{d x} (2 x (4 - x^{2})^{\frac{1}{2}})

\frac{2 ( - 2 x ^{2} + 4 )}{( 4 - x ^{2} ) ^{\frac{1}{2}}}

Solution steps

\frac{d}{d x} (2 x (4 - x^{2})^{\frac{1}{2}})

Take the constant out:

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

= 2 \frac{d}{d x} (x (4 - x^{2})^{\frac{1}{2}})

Apply the Product Rule:

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

f = x, g = (4 - x^{2})^{\frac{1}{2}}

= 2 (\frac{d x}{d x} (4 - x^{2})^{\frac{1}{2}} + \frac{d}{d x} ((4 - x^{2})^{\frac{1}{2}}) x)

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

\frac{d}{d x} ((4 - x^{2})^{\frac{1}{2}}) = - \frac{x}{( 4 - x ^{2} ) ^{\frac{1}{2}}}

= 2 (1 \cdot (4 - x^{2})^{\frac{1}{2}} + (- \frac{x}{( 4 - x ^{2} ) ^{\frac{1}{2}}}) x)

Simplify

2 (1 \cdot (4 - x^{2})^{\frac{1}{2}} + (- \frac{x}{( 4 - x ^{2} ) ^{\frac{1}{2}}}) x) : \frac{2 ( - 2 x ^{2} + 4 )}{( 4 - x ^{2} ) ^{\frac{1}{2}}}

= \frac{2 ( - 2 x ^{2} + 4 )}{( 4 - x ^{2} ) ^{\frac{1}{2}}}

What is the derivative of 2x(4-x^2^{1/2}) ?
The derivative of 2x(4-x^2^{1/2}) is (2(-2x^2+4))/((4-x^2)^{1/2)}
What is the first derivative of 2x(4-x^2^{1/2}) ?
The first derivative of 2x(4-x^2^{1/2}) is (2(-2x^2+4))/((4-x^2)^{1/2)}