Solve 4a^2b*(-ab^2)*5ab-8a^4b^4=

$$4 a ^ { 2 } b \cdot ( - a b ^ { 2 } ) \cdot 5 a b - 8 a ^ { 4 } b ^ { 4 } =$$

Evaluate

$$-28\left(ab\right)^{4}$$

Solution Steps

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

$$4a^{3}b\left(-a\right)b^{2}\times 5b-8a^{4}b^{4}$$

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

$$4a^{3}b^{3}\left(-a\right)\times 5b-8a^{4}b^{4}$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$4a^{3}b^{4}\left(-a\right)\times 5-8a^{4}b^{4}$$

Multiply 4 and 5 to get 20.

$$20a^{3}b^{4}\left(-a\right)-8a^{4}b^{4}$$

Multiply 20 and -1 to get -20.

$$-20a^{3}b^{4}a-8a^{4}b^{4}$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$-20a^{4}b^{4}-8a^{4}b^{4}$$

Combine -20a^{4}b^{4} and -8a^{4}b^{4} to get -28a^{4}b^{4}.

$$-28a^{4}b^{4}$$

Differentiate w.r.t. a

$$-112a^{3}b^{4}$$

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}.

$$2\left(-20a^{2}b^{4}\right)a^{2-1}+4\left(-8b^{4}\right)a^{4-1}$$

Multiply 2 times 4\times 5b\left(-1\right)ab^{2}ab.

$$\left(-40a^{2}b^{4}\right)a^{2-1}+4\left(-8b^{4}\right)a^{4-1}$$

Subtract 1 from 2.

$$\left(-40a^{2}b^{4}\right)a^{1}+4\left(-8b^{4}\right)a^{4-1}$$

Multiply 4 times -8b^{4}.

$$\left(-40a^{2}b^{4}\right)a^{1}+\left(-32b^{4}\right)a^{4-1}$$

Subtract 1 from 4.

$$\left(-40a^{2}b^{4}\right)a^{1}+\left(-32b^{4}\right)a^{3}$$

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

$$\left(-40a^{2}b^{4}\right)a+\left(-32b^{4}\right)a^{3}$$