Solve 4v^{7}w*3v^-7w^7*6u^9u^-9

$$4 v ^ { 7 } w \cdot 3 v ^ { - 7 } w ^ { 7 } \cdot 6 u ^ { 9 } u ^ { - 9 }$$

Evaluate

$$72w^{8}$$

Solution Steps

Multiply v^{7} and v^{-7} to get 1.

$$4w\times 3w^{7}\times 6u^{9}u^{-9}$$

To multiply powers of the same base, add their exponents. Add 1 and 7 to get 8.

$$4w^{8}\times 3\times 6u^{9}u^{-9}$$

Multiply u^{9} and u^{-9} to get 1.

$$4w^{8}\times 3\times 6$$

Multiply 4 and 3 to get 12.

$$12w^{8}\times 6$$

Multiply 12 and 6 to get 72.

$$72w^{8}$$

Differentiate w.r.t. w

$$576w^{7}$$

Steps Using Definition of a Derivative

Multiply v^{7} and v^{-7} to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}w}(4w\times 3w^{7}\times 6u^{9}u^{-9})$$

To multiply powers of the same base, add their exponents. Add 1 and 7 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}w}(4w^{8}\times 3\times 6u^{9}u^{-9})$$

Multiply u^{9} and u^{-9} to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}w}(4w^{8}\times 3\times 6)$$

Multiply 4 and 3 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}w}(12w^{8}\times 6)$$

Multiply 12 and 6 to get 72.

$$\frac{\mathrm{d}}{\mathrm{d}w}(72w^{8})$$

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

$$8\times 72w^{8-1}$$

Multiply 8 times 72.

$$576w^{8-1}$$

Subtract 1 from 8.

$$576w^{7}$$