Solve 1+4r^{2}s^{6}-120r^3s^7+25r^4s^8=

$$1 + 4 r ^ { 2 } s ^ { 6 } - 120 r ^ { 3 } s ^ { 7 } + 25 r ^ { 4 } s ^ { 8 } =$$

Differentiate w.r.t. r

$$4rs^{6}\left(2+25\left(rs\right)^{2}-90rs\right)$$

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}.

$$2\times 4s^{6}r^{2-1}+3\left(-120s^{7}\right)r^{3-1}+4\times 25s^{8}r^{4-1}$$

Multiply 2 times 4s^{6}.

$$8s^{6}r^{2-1}+3\left(-120s^{7}\right)r^{3-1}+4\times 25s^{8}r^{4-1}$$

Subtract 1 from 2.

$$8s^{6}r^{1}+3\left(-120s^{7}\right)r^{3-1}+4\times 25s^{8}r^{4-1}$$

Multiply 3 times -120s^{7}.

$$8s^{6}r^{1}+\left(-360s^{7}\right)r^{3-1}+4\times 25s^{8}r^{4-1}$$

Subtract 1 from 3.

$$8s^{6}r^{1}+\left(-360s^{7}\right)r^{2}+4\times 25s^{8}r^{4-1}$$

Multiply 3 times -120s^{7}.

$$8s^{6}r^{1}+\left(-360s^{7}\right)r^{2}+100s^{8}r^{4-1}$$

Subtract 1 from 4.

$$8s^{6}r^{1}+\left(-360s^{7}\right)r^{2}+100s^{8}r^{3}$$

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

$$8s^{6}r+\left(-360s^{7}\right)r^{2}+100s^{8}r^{3}$$

Evaluate

$$25r^{4}s^{8}-120r^{3}s^{7}+4r^{2}s^{6}+1$$