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For the following equation, evaluate (dy)/(dx) when x=-1. y=2x^(5)+2x^(3) Answer:

For the following equation, evaluate dydx \frac{d y}{d x} when x=−1 x=-1 .\newliney=2x5+2x3 y=2 x^{5}+2 x^{3} \newlineAnswer:

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Q. For the following equation, evaluate dydx \frac{d y}{d x} when x=−1 x=-1 .\newliney=2x5+2x3 y=2 x^{5}+2 x^{3} \newlineAnswer:
  1. Apply Power Rule: To find the derivative of yy with respect to xx, we need to apply the power rule of differentiation, which states that the derivative of xnx^n with respect to xx is n⋅x(n−1)n\cdot x^{(n-1)}.
  2. Differentiate 2x52x^5: Differentiate the first term 2x52x^{5} using the power rule: The derivative is 5×2x5−1=10x45 \times 2x^{5-1} = 10x^{4}.
  3. Differentiate 2x32x^3: Differentiate the second term 2x32x^{3} using the power rule: The derivative is 3×2x3−1=6x23 \times 2x^{3-1} = 6x^{2}.
  4. Combine Derivatives: Combine the derivatives of both terms to get the overall derivative dydx\frac{dy}{dx}: dydx=10x4+6x2\frac{dy}{dx} = 10x^{4} + 6x^{2}.
  5. Substitute x=−1x = -1: Now, substitute x=−1x = -1 into the derivative to evaluate dydx\frac{dy}{dx} at x=−1x = -1: dydx=10(−1)4+6(−1)2\frac{dy}{dx} = 10(-1)^{4} + 6(-1)^{2}.
  6. Calculate Powers: Calculate the powers of −1-1: (−1)4=1(-1)^{4} = 1 and (−1)2=1(-1)^{2} = 1.
  7. Substitute Values: Substitute the values back into the expression: (dydx)=10(1)+6(1)=10+6(\frac{dy}{dx}) = 10(1) + 6(1) = 10 + 6.
  8. Find Final Value: Add the numbers to find the value of (dy)/(dx)(dy)/(dx) when x=−1x = -1: (dy)/(dx)=10+6=16(dy)/(dx) = 10 + 6 = 16.

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