Find the Derivative - d/dx y=5x^8-7x^5+2/3x^3+x-p

Problem

d()/d(x)*(5*x8−7*x5+2/3*x3+x−p)

Solution

1. Identify the expression as a polynomial with respect to x where p is treated as a constant.

2. Apply the power rule, which states that d(xn)/d(x)=n*x(n−1) to each term of the expression.

3. Differentiate the first term: (d(5)*x8)/d(x)=5⋅8*x7=40*x7

4. Differentiate the second term: (d(−)*7*x5)/d(x)=−7⋅5*x4=−35*x4

5. Differentiate the third term: (d(2/3)*x3)/d(x)=2/3⋅3*x2=2*x2

6. Differentiate the linear term: d(x)/d(x)=1

7. Differentiate the constant term: (d(−)*p)/d(x)=0

8. Combine the results to find the final derivative.

Final Answer

d()/d(x)*(5*x8−7*x5+2/3*x3+x−p)=40*x7−35*x4+2*x2+1

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