Solve x^-3/4x-x/6x^5

$$\frac { x ^ { - 3 } } { 4 x } - \frac { x } { 6 x ^ { 5 } }$$

Evaluate

$$\frac{1}{12x^{4}}$$

Solution Steps

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

$$\frac{1}{4x^{4}}-\frac{x}{6x^{5}}$$

Cancel out x in both numerator and denominator.

$$\frac{1}{4x^{4}}-\frac{1}{6x^{4}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4x^{4} and 6x^{4} is 12x^{4}. Multiply \frac{1}{4x^{4}} times \frac{3}{3}. Multiply \frac{1}{6x^{4}} times \frac{2}{2}.

$$\frac{3}{12x^{4}}-\frac{2}{12x^{4}}$$

Since \frac{3}{12x^{4}} and \frac{2}{12x^{4}} have the same denominator, subtract them by subtracting their numerators. Subtract 2 from 3 to get 1.

$$\frac{1}{12x^{4}}$$

Differentiate w.r.t. x

$$-\frac{1}{3x^{5}}$$

Steps Using Chain Rule

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{4x^{4}}-\frac{x}{6x^{5}})$$

Cancel out x in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{4x^{4}}-\frac{1}{6x^{4}})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4x^{4} and 6x^{4} is 12x^{4}. Multiply \frac{1}{4x^{4}} times \frac{3}{3}. Multiply \frac{1}{6x^{4}} times \frac{2}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3}{12x^{4}}-\frac{2}{12x^{4}})$$

Since \frac{3}{12x^{4}} and \frac{2}{12x^{4}} have the same denominator, subtract them by subtracting their numerators. Subtract 2 from 3 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{12x^{4}})$$

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).

$$-\left(12x^{4}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(12x^{4})$$

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}.

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

Simplify.

$$-48x^{3}\times \left(12x^{4}\right)^{-2}$$