Given that y=2x^5+(6)/(sqrt (x)),xgt 0 find in their simplest form (a) (dy)/(dx) (b) int ydx

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Given that y=2x^5+(6)/(sqrt (x)),xgt 0 find in their simplest form (a) (dy)/(dx) (b) int ydx

Answer 1

(a) \( 10x^4 - \frac{3}{x^{1.5}} \) (b) \( \frac{1}{3}x^6 + 12x^{0.5} + c \)

Answer 2

(a) First, we determine the derivative of the function \( y = 2x^5 + \frac{6} {x^{0.5}} \). This involves applying the power rule for differentiation, which states that the derivative of \( x^n \) is \( nx^{n−1} \). The derivative of \( 2x^5 \) is \( 10x^4 \) and the derivative of \( \frac{6} {x^{0.5}} \) is \( -\frac{3}{x^{1.5}} \). (b) Next, we calculate the integral of the function \( y \). The integration rule states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \) provided \( n ≠ -1 \). Finding the integral of \( 2x^5 \) gives \( \frac{2}{6}x^6 = \frac{1}{3}x^6 \), and the integral of \( \frac{6}{x^{0.5}} \) is \( 12x^{0.5} \). It leaves us to add a constant of integration, which we denote as \( c \).

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Given that y=2x^5+(6)/(sqrt (x)),xgt 0 find in their simplest form (a) (dy)/(dx) (b) int ydx

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