Solve (2ab^{2})(4a^2b^3)-(10a^3b)(6b^4)

$$( 2 a b ^ { 2 } ) ( 4 a ^ { 2 } b ^ { 3 } ) - ( 10 a ^ { 3 } b ) ( 6 b ^ { 4 } )$$

Evaluate

$$-52a^{3}b^{5}$$

Solution Steps

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

$$2a^{3}b^{2}\times 4b^{3}-10a^{3}b\times 6b^{4}$$

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

$$2a^{3}b^{5}\times 4-10a^{3}b\times 6b^{4}$$

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

$$2a^{3}b^{5}\times 4-10a^{3}b^{5}\times 6$$

Multiply 2 and 4 to get 8.

$$8a^{3}b^{5}-10a^{3}b^{5}\times 6$$

Multiply 10 and 6 to get 60.

$$8a^{3}b^{5}-60a^{3}b^{5}$$

Combine 8a^{3}b^{5} and -60a^{3}b^{5} to get -52a^{3}b^{5}.

$$-52a^{3}b^{5}$$

Differentiate w.r.t. a

$$-156a^{2}b^{5}$$

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}.

$$8a^{2}b^{5}a^{1-1}+3\left(-60b^{5}\right)a^{3-1}$$

Subtract 1 from 1.

$$8a^{2}b^{5}a^{0}+3\left(-60b^{5}\right)a^{3-1}$$

Multiply 3 times -10\times 6bb^{4}.

$$8a^{2}b^{5}a^{0}+\left(-180b^{5}\right)a^{3-1}$$

Subtract 1 from 3.

$$8a^{2}b^{5}a^{0}+\left(-180b^{5}\right)a^{2}$$

For any term t except 0, t^{0}=1.

$$8a^{2}b^{5}\times 1+\left(-180b^{5}\right)a^{2}$$

For any term t, t\times 1=t and 1t=t.

$$8a^{2}b^{5}+\left(-180b^{5}\right)a^{2}$$