Solve frac{(64x^3)^-frac{1{3}}}{4x^-frac{1{4}}}

$$\frac { ( 64 x ^ { 3 } ) ^ { - \frac { 1 } { 3 } } } { 4 x ^ { - \frac { 1 } { 4 } } }$$

Evaluate

$$\frac{1}{16x^{\frac{3}{4}}}$$

Solution Steps

Expand \left(64x^{3}\right)^{-\frac{1}{3}}.

$$\frac{64^{-\frac{1}{3}}\left(x^{3}\right)^{-\frac{1}{3}}}{4x^{-\frac{1}{4}}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -\frac{1}{3} to get -1.

$$\frac{64^{-\frac{1}{3}}x^{-1}}{4x^{-\frac{1}{4}}}$$

Calculate 64 to the power of -\frac{1}{3} and get \frac{1}{4}.

$$\frac{\frac{1}{4}x^{-1}}{4x^{-\frac{1}{4}}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\frac{1}{4}}{4x^{\frac{3}{4}}}$$

Express \frac{\frac{1}{4}}{4x^{\frac{3}{4}}} as a single fraction.

$$\frac{1}{4\times 4x^{\frac{3}{4}}}$$

Multiply 4 and 4 to get 16.

$$\frac{1}{16x^{\frac{3}{4}}}$$

Differentiate w.r.t. x

$$-\frac{3}{64x^{\frac{7}{4}}}$$

Steps Using Chain Rule

Expand \left(64x^{3}\right)^{-\frac{1}{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{64^{-\frac{1}{3}}\left(x^{3}\right)^{-\frac{1}{3}}}{4x^{-\frac{1}{4}}})$$

To raise a power to another power, multiply the exponents. Multiply 3 and -\frac{1}{3} to get -1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{64^{-\frac{1}{3}}x^{-1}}{4x^{-\frac{1}{4}}})$$

Calculate 64 to the power of -\frac{1}{3} and get \frac{1}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{4}x^{-1}}{4x^{-\frac{1}{4}}})$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{4}}{4x^{\frac{3}{4}}})$$

Express \frac{\frac{1}{4}}{4x^{\frac{3}{4}}} as a single fraction.

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

Multiply 4 and 4 to get 16.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{16x^{\frac{3}{4}}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(16x^{\frac{3}{4}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(16x^{\frac{3}{4}})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-\left(16x^{\frac{3}{4}}\right)^{-2}\times \frac{3}{4}\times 16x^{\frac{3}{4}-1}$$

Simplify.

$$-12x^{-\frac{1}{4}}\times \left(16x^{\frac{3}{4}}\right)^{-2}$$