To find the first and second derivatives of the given function y = 7x - 5 - x^5 with respect to x, you have to use the power rule for differentiation. The power rule states that the derivative of x^n with respect to x is n*x^(n-1).
To find the first derivative, we need to apply the power rule to each term of the function:
1. For the first term, 7x, the exponent is 1, so applying the power rule gives us 1*7*x^(1-1) = 7.
2. The second term is negative 5, which is a constant. The derivative of a constant is 0.
3. For the third term, -x^5, the exponent is 5, so applying the power rule gives us -5*x^(5-1) = -5x^4.
Summing these up, the first derivative of y = 7x - 5 - x^5 is therefore 7 - 5x^4.
To find the second derivative, we again apply the power rule to each term of the first derivative:
1. The first term 7 becomes 0, because it is a constant and the derivative of a constant is zero.
2. For the second term, -5x^4, the power is 4. According to the power rule, this term's derivative is -20*x^(4-1) = -20x^3.
Summing these up, we see that the second derivative of y = 7x - 5 - x^5 is -20x^3.
So, the first derivative is 7 - 5x^4, and the second derivative is -20x^3.