What is the d/(d{r)}({r}sqrt(1-{r)^2}) ?

The d/(d{r)}({r}sqrt(1-{r)^2}) is sqrt(1-{r)^2}-({r}^2)/(sqrt(1-{r)^2)}

What is the first d/(d{r)}({r}sqrt(1-{r)^2}) ?

The first d/(d{r)}({r}sqrt(1-{r)^2}) is sqrt(1-{r)^2}-({r}^2)/(sqrt(1-{r)^2)}

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\frac{d}{d r} (r 1 - r^{2})

1 - r^{2} - \frac{r ^{2}}{1 - r ^{2}}

Solution steps

\frac{d}{d r} (r 1 - r^{2})

Apply the Product Rule:

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

f = r, g = 1 - r^{2}

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

\frac{d}{d r} (r) = 1

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

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

Simplify

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

= 1 - r^{2} - \frac{r ^{2}}{1 - r ^{2}}

What is the d/(d{r)}({r}sqrt(1-{r)^2}) ?
The d/(d{r)}({r}sqrt(1-{r)^2}) is sqrt(1-{r)^2}-({r}^2)/(sqrt(1-{r)^2)}
What is the first d/(d{r)}({r}sqrt(1-{r)^2}) ?
The first d/(d{r)}({r}sqrt(1-{r)^2}) is sqrt(1-{r)^2}-({r}^2)/(sqrt(1-{r)^2)}