Solve 23p^{3}q^5-4p^2q^3*3pq^2

$$23 p ^ { 3 } q ^ { 5 } - 4 p ^ { 2 } q ^ { 3 } \times 3 p q ^ { 2 }$$

Evaluate

$$11p^{3}q^{5}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$23p^{3}q^{5}-4p^{3}q^{3}\times 3q^{2}$$

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$23p^{3}q^{5}-4p^{3}q^{5}\times 3$$

Multiply 4 and 3 to get 12.

$$23p^{3}q^{5}-12p^{3}q^{5}$$

Combine 23p^{3}q^{5} and -12p^{3}q^{5} to get 11p^{3}q^{5}.

$$11p^{3}q^{5}$$

Differentiate w.r.t. p

$$33p^{2}q^{5}$$

Steps Using Derivative Rule for Sum

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$3\times 23q^{5}p^{3-1}+2\left(-12pq^{5}\right)p^{2-1}$$

Multiply 3 times 23q^{5}.

$$69q^{5}p^{3-1}+2\left(-12pq^{5}\right)p^{2-1}$$

Subtract 1 from 3.

$$69q^{5}p^{2}+2\left(-12pq^{5}\right)p^{2-1}$$

Multiply 2 times -4\times 3q^{3}pq^{2}.

$$69q^{5}p^{2}+\left(-24pq^{5}\right)p^{2-1}$$

Subtract 1 from 2.

$$69q^{5}p^{2}+\left(-24pq^{5}\right)p^{1}$$

For any term t, t^{1}=t.

$$69q^{5}p^{2}+\left(-24pq^{5}\right)p$$