What is the derivative of (x(sqrt(-4x^2+1)+1))/4 ?

The derivative of (x(sqrt(-4x^2+1)+1))/4 is 1/4 ((-8x^2+1)/(sqrt(-4x^2+1))+1)

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

The first derivative of (x(sqrt(-4x^2+1)+1))/4 is 1/4 ((-8x^2+1)/(sqrt(-4x^2+1))+1)

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derivative of

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

\frac{1}{4} (\frac{- 8 x ^{2} + 1}{- 4 x ^{2} + 1} + 1)

Solution steps

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

Take the constant out:

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

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

Apply the Product Rule:

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

f = x, g = - 4 x^{2} + 1 + 1

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

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

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

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

Simplify

\frac{1}{4} (1 \cdot (- 4 x^{2} + 1 + 1) + (- \frac{4 x}{- 4 x ^{2} + 1}) x) : \frac{1}{4} (\frac{- 8 x ^{2} + 1}{- 4 x ^{2} + 1} + 1)

= \frac{1}{4} (\frac{- 8 x ^{2} + 1}{- 4 x ^{2} + 1} + 1)

x \to 0 lim (\frac{4 + x - 4 - x}{4 x})

x \to - \frac{π}{2} lim (- tan (0.5 x))

\frac{\partial}{\partial y} (3 x e^{y})

\frac{\partial}{\partial x} (- e^{x})

\int_{0}^{\frac{π}{2}} x^{2} sin (x) d x

What is the derivative of (x(sqrt(-4x^2+1)+1))/4 ?
The derivative of (x(sqrt(-4x^2+1)+1))/4 is 1/4 ((-8x^2+1)/(sqrt(-4x^2+1))+1)
What is the first derivative of (x(sqrt(-4x^2+1)+1))/4 ?
The first derivative of (x(sqrt(-4x^2+1)+1))/4 is 1/4 ((-8x^2+1)/(sqrt(-4x^2+1))+1)