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

Problem

d()/d(x)*(5*x^8−7*x^5+2/3*x^3+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(x^n)/d(x)=n*x^(n−1) to each term of the expression.

3. Differentiate the first term: (d(5)*x^8)/d(x)=5⋅8*x^7=40*x^7

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

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

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*x^8−7*x^5+2/3*x^3+x−p)=40*x^7−35*x^4+2*x^2+1

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