Solve x^-3/x^4*x^-7/x^-2

$$\frac { x ^ { - 3 } } { x ^ { 4 } } \cdot \frac { x ^ { - 7 } } { x ^ { - 2 } }$$

Evaluate

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

Solution Steps

Rewrite x^{4} as x^{-3}x^{7}. Cancel out x^{-3} in both numerator and denominator.

$$\frac{1}{x^{7}}\times \frac{x^{-7}}{x^{-2}}$$

Rewrite x^{-2} as x^{-7}x^{5}. Cancel out x^{-7} in both numerator and denominator.

$$\frac{1}{x^{7}}\times \frac{1}{x^{5}}$$

Multiply \frac{1}{x^{7}} times \frac{1}{x^{5}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{x^{7}x^{5}}$$

To multiply powers of the same base, add their exponents. Add 7 and 5 to get 12.

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

Differentiate w.r.t. x

$$-\frac{12}{x^{13}}$$

Steps Using Chain Rule

Rewrite x^{4} as x^{-3}x^{7}. Cancel out x^{-3} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{7}}\times \frac{x^{-7}}{x^{-2}})$$

Rewrite x^{-2} as x^{-7}x^{5}. Cancel out x^{-7} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{7}}\times \frac{1}{x^{5}})$$

Multiply \frac{1}{x^{7}} times \frac{1}{x^{5}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{7}x^{5}})$$

To multiply powers of the same base, add their exponents. Add 7 and 5 to get 12.

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

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).

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

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}.

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

Simplify.

$$-12x^{11}\left(x^{12}\right)^{-2}$$