Solve x^5*x^-4divx^7

$${ x }^{ 5 } \times { x }^{ -4 } \div { x }^{ 7 }$$

Evaluate

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

Solution Steps

To multiply powers of the same base, add their exponents. Add 5 and -4 to get 1.

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

Rewrite x^{7} as x^{1}x^{6}. Cancel out x^{1} in both numerator and denominator.

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

Differentiate w.r.t. x

$$-\frac{6}{x^{7}}$$

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add 5 and -4 to get 1.

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

Rewrite x^{7} as x^{1}x^{6}. Cancel out x^{1} in both numerator and denominator.

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

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).

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

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}.

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

Simplify.

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