Solve 8yx^-5*3x^5y^2=

$$8 y x ^ { - 5 } \cdot 3 x ^ { 5 } y ^ { 2 } =$$

Evaluate

$$24y^{3}$$

Steps Using Commutative Property of Multiplication

Use the rules of exponents to simplify the expression.

$$8^{1}y^{1}x^{-5}\times 3^{1}y^{2}x^{5}$$

Use the Commutative Property of Multiplication.

$$8^{1}\times 3^{1}y^{1}y^{2}x^{-5}x^{5}$$

To multiply powers of the same base, add their exponents.

$$8^{1}\times 3^{1}y^{1+2}x^{-5+5}$$

Add the exponents 1 and 2.

$$8^{1}\times 3^{1}y^{3}x^{-5+5}$$

Add the exponents -5 and 5.

$$8^{1}\times 3^{1}y^{3}x^{0}$$

For any number a except 0, a^{0}=1.

$$8^{1}\times 3^{1}y^{3}$$

Multiply 8 times 3.

$$24y^{3}$$

Differentiate w.r.t. y

$$72y^{2}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(8y^{3}x^{-5}\times 3x^{5})$$

Multiply x^{-5} and x^{5} to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}y}(8y^{3}\times 3)$$

Multiply 8 and 3 to get 24.

$$\frac{\mathrm{d}}{\mathrm{d}y}(24y^{3})$$

The derivative of ax^{n} is nax^{n-1}.

$$3\times 24y^{3-1}$$

Multiply 3 times 24.

$$72y^{3-1}$$

Subtract 1 from 3.

$$72y^{2}$$