What is the derivative of y=(x+x^2)^{1/2} ?

The derivative of y=(x+x^2)^{1/2} is (1+2x)/(2(x+x^2)^{1/2)}

What is the first derivative of y=(x+x^2)^{1/2} ?

The first derivative of y=(x+x^2)^{1/2} is (1+2x)/(2(x+x^2)^{1/2)}

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derivative

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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