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How do you find the derivative of #x+sqrt(x)#?

Answer 1

Emma Aldridge

#color(blue)(1+1/2x^(-1/2))# or #color(blue)(1+1/(2sqrt(x)))#

#f(x)=x+sqrt(x)#

Rewriting as:

#f(x)=x+x^(1/2)#

We can differentiate this using the power rule:

The Power Rules states that:

#dy/dx(ax^n)=nax^(n-1)#

#dy/dx# is distributive over the sum, so:

#dy/dx(ax^2+bx+c)=dy/dx(ax^2)+dy/dx(bx)+dy/dx(c)#

Differentiating #f(x)#

#dy/dx(x+x^(1/2))=1*x^(1-1)+1/2*x^(1/2-1)#

#=x^0+1/2x^(-1/2)=color(blue)(1+1/2x^(-1/2))# or #color(blue)(1+1/(2sqrt(x)))#

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Answer 2

Isaiah Allen

To find the derivative of (x + \sqrt{x}), you can use the sum rule and the power rule of differentiation. The derivative of (x) with respect to (x) is (1), and the derivative of (\sqrt{x}) with respect to (x) is (\frac{1}{2\sqrt{x}}). Therefore, the derivative of (x + \sqrt{x}) with respect to (x) is (1 + \frac{1}{2\sqrt{x}}).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.