Question

Y=2x^4-5x^-4+3x+4 y'(1)

Answer 1

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Answer 2

To find the derivative of the function Y = 2x^4 - 5x^(-4) + 3x + 4 and evaluate it at x = 1, we can differentiate each term separately using the power rule and sum the results.

Let's calculate the derivative step by step:

1. Derivative of 2x^4: The power rule states that if we have a term of the form ax^n, its derivative is given by (a * n) * x^(n - 1). Applying this rule, we get: d/dx (2x^4) = 2 * 4 * x^(4 - 1) = 8x^3.

2. Derivative of -5x^(-4): Applying the power rule again, we have: d/dx (-5x^(-4)) = -5 * (-4) * x^(-4 - 1) = 20x^(-5).

3. Derivative of 3x: The derivative of a linear term ax is simply a since the power is 1. Therefore: d/dx (3x) = 3.

4. Derivative of 4: The derivative of a constant term is always zero: d/dx (4) = 0.

Now, we can sum up the derivatives we obtained: Y'(x) = 8x^3 + 20x^(-5) + 3 + 0.

Finally, to evaluate Y'(1), we substitute x = 1 into the derivative expression: Y'(1) = 8(1)^3 + 20(1)^(-5) + 3 + 0 = 8(1) + 20(1) + 3 = 8 + 20 + 3 = 31.

Therefore, Y'(1) = 31.