Solve 4w^{9}u*4w^-7u^-1*4x^-8x^8

$$4 w ^ { 9 } u \cdot 4 w ^ { - 7 } u ^ { - 1 } \cdot 4 x ^ { - 8 } x ^ { 8 }$$

Evaluate

$$64w^{2}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 9 and -7 to get 2.

$$4w^{2}u\times 4u^{-1}\times 4x^{-8}x^{8}$$

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

$$4w^{2}\times 4\times 4x^{-8}x^{8}$$

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

$$4w^{2}\times 4\times 4$$

Multiply 4 and 4 to get 16.

$$16w^{2}\times 4$$

Multiply 16 and 4 to get 64.

$$64w^{2}$$

Differentiate w.r.t. w

$$128w$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 9 and -7 to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}w}(4w^{2}u\times 4u^{-1}\times 4x^{-8}x^{8})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}w}(4w^{2}\times 4\times 4x^{-8}x^{8})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}w}(4w^{2}\times 4\times 4)$$

Multiply 4 and 4 to get 16.

$$\frac{\mathrm{d}}{\mathrm{d}w}(16w^{2}\times 4)$$

Multiply 16 and 4 to get 64.

$$\frac{\mathrm{d}}{\mathrm{d}w}(64w^{2})$$

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

$$2\times 64w^{2-1}$$

Multiply 2 times 64.

$$128w^{2-1}$$

Subtract 1 from 2.

$$128w^{1}$$

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

$$128w$$