Solve 10m^{3}m^2+15m^4m-20m^2m^3

$$10 m ^ { 3 } m ^ { 2 } + 15 m ^ { 4 } m - 20 m ^ { 2 } m ^ { 3 }$$

Evaluate

$$5m^{5}$$

Solution Steps

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

$$10m^{5}+15m^{4}m-20m^{2}m^{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$10m^{5}+15m^{5}-20m^{2}m^{3}$$

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

$$10m^{5}+15m^{5}-20m^{5}$$

Combine 10m^{5} and 15m^{5} to get 25m^{5}.

$$25m^{5}-20m^{5}$$

Combine 25m^{5} and -20m^{5} to get 5m^{5}.

$$5m^{5}$$

Differentiate w.r.t. m

$$25m^{4}$$

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}.

$$3\times 10m^{2}m^{3-1}+4\times 15mm^{4-1}+2\left(-20m^{3}\right)m^{2-1}$$

Multiply 3 times 10m^{2}.

$$30m^{2}m^{3-1}+4\times 15mm^{4-1}+2\left(-20m^{3}\right)m^{2-1}$$

Subtract 1 from 3.

$$30m^{2}m^{2}+4\times 15mm^{4-1}+2\left(-20m^{3}\right)m^{2-1}$$

Multiply 4 times 15m.

$$30m^{2}m^{2}+60mm^{4-1}+2\left(-20m^{3}\right)m^{2-1}$$

Subtract 1 from 4.

$$30m^{2}m^{2}+60mm^{3}+2\left(-20m^{3}\right)m^{2-1}$$

Multiply 4 times 15m.

$$30m^{2}m^{2}+60mm^{3}+\left(-40m^{3}\right)m^{2-1}$$

Subtract 1 from 2.

$$30m^{2}m^{2}+60mm^{3}+\left(-40m^{3}\right)m^{1}$$

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

$$30m^{2}m^{2}+60mm^{3}+\left(-40m^{3}\right)m$$