Differentiate the following with respect to P:
A)567K b)897PK c)23P^-1 d)2KP^6+4P^5+P^4+6K^3-K^2-329K-10,000
There is no P in the first (A) term. If K is a constant, the derivative with respect to P is 0.
Yse the rules of polynomial differentiation for the others. We will be glad to critique your work.
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To differentiate with respect to a variable (in this case, P), we can use the power rule for differentiation. The power rule states that if we have a term of the form ax^n, where a and n are constants, the derivative with respect to x (in our case, P) is given by nax^(n-1).
A) Let's differentiate 567K with respect to P. Here, K is a constant.
Since 567 is a constant, its derivative with respect to P is 0.
Therefore, the derivative with respect to P of 567K is 0.
B) Let's differentiate 897PK with respect to P.
Since 897 and K are constants, their derivatives with respect to P are both 0.
Using the power rule, we differentiate P with respect to P to get 1.
Therefore, the derivative with respect to P of 897PK is 897K.
C) Let's differentiate 23P^(-1) with respect to P.
Using the power rule, we differentiate P^(-1) with respect to P to get (-1)P^(-2).
Therefore, the derivative with respect to P of 23P^(-1) is -23P^(-2).
D) Let's differentiate 2KP^6 + 4P^5 + P^4 + 6K^3 - K^2 - 329K - 10,000 with respect to P.
Using the power rule, we differentiate each term as follows:
- The derivative of 2KP^6 with respect to P is 2K(6)P^(6-1) = 12KP^5.
- The derivative of 4P^5 with respect to P is 4(5)P^(5-1) = 20P^4.
- The derivative of P^4 with respect to P is 4P^(4-1) = 4P^3.
- The derivative of 6K^3 with respect to P is 0, as K is a constant.
- The derivative of -K^2 with respect to P is 0, as K is a constant.
- The derivative of -329K with respect to P is 0, as K is a constant.
- The derivative of -10,000 with respect to P is 0, as it is a constant.
Adding up all the derivatives, we get: 12KP^5 + 20P^4 + 4P^3.
Therefore, the derivative with respect to P of 2KP^6 + 4P^5 + P^4 + 6K^3 - K^2 - 329K - 10,000 is 12KP^5 + 20P^4 + 4P^3.