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Derivative of a sum involving power, reciprocal, and square root terms

Let f(x)=6x5−3x−2x.f(x) = 6x^5 - \dfrac{3}{x} - 2\sqrt{x}. Rewrite each term using exponent notation, then use the constant multiple and sum/difference rules together with the power rule to find f′(x)f'(x).

Hint

First rewrite 3x\dfrac{3}{x} as 3x−13x^{-1} and 2x2\sqrt{x} as 2x1/22x^{1/2}. Then differentiate term by term, applying the power rule to each piece and keeping the constant multiples and signs attached.

Step-by-step solution

Rewrite f(x)=6x5−3x−1−2x1/2f(x) = 6x^5 - 3x^{-1} - 2x^{1/2} so that every term is in the form c xnc\,x^n. This makes it possible to apply the power rule directly to each piece.

By the sum/difference and constant multiple rules, differentiate term by term: ddx[6x5]=6⋅5x4=30x4\frac{d}{dx}[6x^5] = 6 \cdot 5x^4 = 30x^4, ddx[−3x−1]=−3⋅(−1)x−2=3x−2\frac{d}{dx}[-3x^{-1}] = -3 \cdot (-1)x^{-2} = 3x^{-2}, and ddx[−2x1/2]=−2⋅12x−1/2=−x−1/2\frac{d}{dx}[-2x^{1/2}] = -2 \cdot \frac{1}{2}x^{-1/2} = -x^{-1/2}. Combining these gives f′(x)=30x4+3x2−1x.f'(x) = 30x^4 + \frac{3}{x^2} - \frac{1}{\sqrt{x}}.

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