Solve (3/5x)^-1

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

Evaluate

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

Solution Steps

To raise \frac{3}{5x} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

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

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

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

Express \frac{\frac{1}{3}}{\frac{1}{5}x^{-1}} as a single fraction.

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

Multiply 3 and \frac{1}{5} to get \frac{3}{5}.

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

Differentiate w.r.t. x

$$\frac{5}{3} = 1\frac{2}{3} = 1.6666666666666667$$

Steps Using Chain Rule

To raise \frac{3}{5x} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

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

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

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

Express \frac{\frac{1}{3}}{\frac{1}{5}x^{-1}} as a single fraction.

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

Multiply 3 and \frac{1}{5} to get \frac{3}{5}.

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

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).

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

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}.

$$-\left(\frac{3}{5}\times \frac{1}{x}\right)^{-2}\left(-1\right)\times \frac{3}{5}x^{-1-1}$$

Simplify.

$$\frac{3}{5}x^{-2}\times \left(\frac{3}{5}\times \frac{1}{x}\right)^{-2}$$