Solve 3v^{5}x^8*5x^-8w^-1*5v^-3w

$$3 v ^ { 5 } x ^ { 8 } \cdot 5 x ^ { - 8 } w ^ { - 1 } \cdot 5 v ^ { - 3 } w$$

Evaluate

$$75v^{2}$$

Solution Steps

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

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

Multiply x^{8} and x^{-8} to get 1.

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

Multiply w^{-1} and w to get 1.

$$3v^{2}\times 5\times 5$$

Multiply 3 and 5 to get 15.

$$15v^{2}\times 5$$

Multiply 15 and 5 to get 75.

$$75v^{2}$$

Differentiate w.r.t. v

$$150v$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}v}(3v^{2}x^{8}\times 5x^{-8}w^{-1}\times 5w)$$

Multiply x^{8} and x^{-8} to get 1.

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

Multiply w^{-1} and w to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}v}(3v^{2}\times 5\times 5)$$

Multiply 3 and 5 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}v}(15v^{2}\times 5)$$

Multiply 15 and 5 to get 75.

$$\frac{\mathrm{d}}{\mathrm{d}v}(75v^{2})$$

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

$$2\times 75v^{2-1}$$

Multiply 2 times 75.

$$150v^{2-1}$$

Subtract 1 from 2.

$$150v^{1}$$

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

$$150v$$