Solve (5w^4)(2w^-3)(0.5w^5)(1/5w^-3)

$$( 5 w ^ { 4 } ) ( 2 w ^ { - 3 } ) ( 0.5 w ^ { 5 } ) ( \frac { 1 } { 5 } w ^ { - 3 } )$$

Evaluate

$$w^{3}$$

Solution Steps

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

$$5w^{1}\times 2\times 0.5w^{5}\times \frac{1}{5}w^{-3}$$

To multiply powers of the same base, add their exponents. Add 1 and 5 to get 6.

$$5w^{6}\times 2\times 0.5\times \frac{1}{5}w^{-3}$$

To multiply powers of the same base, add their exponents. Add 6 and -3 to get 3.

$$5w^{3}\times 2\times 0.5\times \frac{1}{5}$$

Multiply 5 and 2 to get 10.

$$10w^{3}\times 0.5\times \frac{1}{5}$$

Multiply 10 and 0.5 to get 5.

$$5w^{3}\times \frac{1}{5}$$

Multiply 5 and \frac{1}{5} to get 1.

$$w^{3}$$

Differentiate w.r.t. w

$$3w^{2}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}w}(5w^{1}\times 2\times 0.5w^{5}\times \frac{1}{5}w^{-3})$$

To multiply powers of the same base, add their exponents. Add 1 and 5 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}w}(5w^{6}\times 2\times 0.5\times \frac{1}{5}w^{-3})$$

To multiply powers of the same base, add their exponents. Add 6 and -3 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}w}(5w^{3}\times 2\times 0.5\times \frac{1}{5})$$

Multiply 5 and 2 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}w}(10w^{3}\times 0.5\times \frac{1}{5})$$

Multiply 10 and 0.5 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}w}(5w^{3}\times \frac{1}{5})$$

Multiply 5 and \frac{1}{5} to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}w}(w^{3})$$

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

$$3w^{3-1}$$

Subtract 1 from 3.

$$3w^{2}$$