Solve 6x/9y*3y/2x^5

$$\frac { 6 x } { 9 y } \cdot \frac { 3 y } { 2 x ^ { 5 } }$$

Evaluate

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

Solution Steps

Cancel out 3 in both numerator and denominator.

$$\frac{2x}{3y}\times \frac{3y}{2x^{5}}$$

Multiply \frac{2x}{3y} times \frac{3y}{2x^{5}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2x\times 3y}{3y\times 2x^{5}}$$

Cancel out 2\times 3xy in both numerator and denominator.

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

Differentiate w.r.t. x

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

Steps Using Chain Rule

Cancel out 3 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x}{3y}\times \frac{3y}{2x^{5}})$$

Multiply \frac{2x}{3y} times \frac{3y}{2x^{5}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x\times 3y}{3y\times 2x^{5}})$$

Cancel out 2\times 3xy in both numerator and denominator.

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

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).

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

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}.

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

Simplify.

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