Solve 1/y/2x*1/2x/frac{1{y}}

$$\frac { \frac { 1 } { y } } { 2 x } \cdot \frac { \frac { 1 } { 2 x } } { \frac { 1 } { y } }$$

Evaluate

$$\frac{1}{4x^{2}}$$

Solution Steps

Express \frac{\frac{1}{y}}{2x} as a single fraction.

$$\frac{1}{y\times 2x}\times \frac{\frac{1}{2x}}{\frac{1}{y}}$$

Divide \frac{1}{2x} by \frac{1}{y} by multiplying \frac{1}{2x} by the reciprocal of \frac{1}{y}.

$$\frac{1}{y\times 2x}\times \frac{y}{2x}$$

Multiply \frac{1}{y\times 2x} times \frac{y}{2x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{y}{y\times 2x\times 2x}$$

Cancel out y in both numerator and denominator.

$$\frac{1}{2\times 2xx}$$

Multiply x and x to get x^{2}.

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

Multiply 2 and 2 to get 4.

$$\frac{1}{4x^{2}}$$

Differentiate w.r.t. x

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

Steps Using Chain Rule

Express \frac{\frac{1}{y}}{2x} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{y\times 2x}\times \frac{\frac{1}{2x}}{\frac{1}{y}})$$

Divide \frac{1}{2x} by \frac{1}{y} by multiplying \frac{1}{2x} by the reciprocal of \frac{1}{y}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{y\times 2x}\times \frac{y}{2x})$$

Multiply \frac{1}{y\times 2x} times \frac{y}{2x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y}{y\times 2x\times 2x})$$

Cancel out y in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2\times 2xx})$$

Multiply x and x to get x^{2}.

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

Multiply 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{4x^{2}})$$

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).

$$-\left(4x^{2}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(4x^{2})$$

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}.

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

Simplify.

$$-8x^{1}\times \left(4x^{2}\right)^{-2}$$

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

$$-8x\times \left(4x^{2}\right)^{-2}$$