Solve 4x^-frac{1{3}}*4x^-4*x^frac{-1{2}}

$$4 x ^ { - \frac { 1 } { 3 } } \cdot 4 x ^ { - 4 } \cdot x ^ { \frac { - 1 } { 2 } }$$

Evaluate

$$\frac{16}{x^{\frac{29}{6}}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -\frac{1}{3} and -4 to get -\frac{13}{3}.

$$4x^{-\frac{13}{3}}\times 4x^{\frac{-1}{2}}$$

Multiply 4 and 4 to get 16.

$$16x^{-\frac{13}{3}}x^{\frac{-1}{2}}$$

Fraction \frac{-1}{2} can be rewritten as -\frac{1}{2} by extracting the negative sign.

$$16x^{-\frac{13}{3}}x^{-\frac{1}{2}}$$

To multiply powers of the same base, add their exponents. Add -\frac{13}{3} and -\frac{1}{2} to get -\frac{29}{6}.

$$16x^{-\frac{29}{6}}$$

Differentiate w.r.t. x

$$-\frac{232}{3x^{\frac{35}{6}}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add -\frac{1}{3} and -4 to get -\frac{13}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{-\frac{13}{3}}\times 4x^{\frac{-1}{2}})$$

Multiply 4 and 4 to get 16.

$$\frac{\mathrm{d}}{\mathrm{d}x}(16x^{-\frac{13}{3}}x^{\frac{-1}{2}})$$

Fraction \frac{-1}{2} can be rewritten as -\frac{1}{2} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}x}(16x^{-\frac{13}{3}}x^{-\frac{1}{2}})$$

To multiply powers of the same base, add their exponents. Add -\frac{13}{3} and -\frac{1}{2} to get -\frac{29}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(16x^{-\frac{29}{6}})$$

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

$$-\frac{29}{6}\times 16x^{-\frac{29}{6}-1}$$

Multiply -\frac{29}{6} times 16.

$$-\frac{232}{3}x^{-\frac{29}{6}-1}$$

Subtract 1 from -\frac{29}{6}.

$$-\frac{232}{3}x^{-\frac{35}{6}}$$