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How do you use the power rule to differentiate #f(x)=1-x^2+3x^3sqrtx#?

Answer 1

Emily Ammerman

#f'(x)=21/2sqrt(x^5)-2x#

differentiate using the #color(blue)"power rule"# on each term.

#color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(d/dx(ax^n)=nax^(n-1))color(white)(2/2)|)))#

Express #3x^3sqrtx=3x^3.x^(1/2)=3x^(7/2)#

#f(x)=1-x^2+3x^(7/2)#

#rArrf'(x)=0-2x+21/2x^(5/2)#

#=21/2sqrt(x^5)-2x#

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Answer 2

Adam Adkins

To use the power rule to differentiate (f(x) = 1 - x^2 + 3x^3\sqrt{x}), follow these steps:

Differentiate each term of the function separately.
For (1), the derivative is (0) since it's a constant.
For (-x^2), apply the power rule: (d/dx(-x^2) = -2x).
For (3x^3\sqrt{x}), apply the power rule: (d/dx(3x^3\sqrt{x}) = 9x^2\sqrt{x} + \frac{3}{2}x^3 \cdot \frac{1}{\sqrt{x}}).
Simplify the derivative: (f'(x) = -2x + 9x^2\sqrt{x} + \frac{3}{2}x^{3/2}).

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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.