Solve x^-3(6/5x)^-1

$$x ^ { - 3 } ( \frac { 6 } { 5 x } ) ^ { - 1 }$$

Evaluate

$$\frac{5}{6x^{2}}$$

Solution Steps

To raise \frac{6}{5x} to a power, raise both numerator and denominator to the power and then divide.

$$x^{-3}\times \frac{6^{-1}}{\left(5x\right)^{-1}}$$

Express x^{-3}\times \frac{6^{-1}}{\left(5x\right)^{-1}} as a single fraction.

$$\frac{x^{-3}\times 6^{-1}}{\left(5x\right)^{-1}}$$

Calculate 6 to the power of -1 and get \frac{1}{6}.

$$\frac{x^{-3}\times \frac{1}{6}}{\left(5x\right)^{-1}}$$

Expand \left(5x\right)^{-1}.

$$\frac{x^{-3}\times \frac{1}{6}}{5^{-1}x^{-1}}$$

Calculate 5 to the power of -1 and get \frac{1}{5}.

$$\frac{x^{-3}\times \frac{1}{6}}{\frac{1}{5}x^{-1}}$$

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

$$\frac{\frac{1}{6}}{\frac{1}{5}x^{2}}$$

Express \frac{\frac{1}{6}}{\frac{1}{5}x^{2}} as a single fraction.

$$\frac{1}{6\times \frac{1}{5}x^{2}}$$

Multiply 6 and \frac{1}{5} to get \frac{6}{5}.

$$\frac{1}{\frac{6}{5}x^{2}}$$

Differentiate w.r.t. x

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

Steps Using Chain Rule

To raise \frac{6}{5x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{-3}\times \frac{6^{-1}}{\left(5x\right)^{-1}})$$

Express x^{-3}\times \frac{6^{-1}}{\left(5x\right)^{-1}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-3}\times 6^{-1}}{\left(5x\right)^{-1}})$$

Calculate 6 to the power of -1 and get \frac{1}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-3}\times \frac{1}{6}}{\left(5x\right)^{-1}})$$

Expand \left(5x\right)^{-1}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-3}\times \frac{1}{6}}{5^{-1}x^{-1}})$$

Calculate 5 to the power of -1 and get \frac{1}{5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-3}\times \frac{1}{6}}{\frac{1}{5}x^{-1}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{6}}{\frac{1}{5}x^{2}})$$

Express \frac{\frac{1}{6}}{\frac{1}{5}x^{2}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{6\times \frac{1}{5}x^{2}})$$

Multiply 6 and \frac{1}{5} to get \frac{6}{5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{\frac{6}{5}x^{2}})$$

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).

$$-\left(\frac{6}{5}x^{2}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6}{5}x^{2})$$

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}.

$$-\left(\frac{6}{5}x^{2}\right)^{-2}\times 2\times \frac{6}{5}x^{2-1}$$

Simplify.

$$-\frac{12}{5}x^{1}\times \left(\frac{6}{5}x^{2}\right)^{-2}$$

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

$$-\frac{12}{5}x\times \left(\frac{6}{5}x^{2}\right)^{-2}$$