Solve (x^10)^-7/x^9

$$\frac { ( x ^ { 10 } ) ^ { - 7 } } { x ^ { 9 } }$$

Evaluate

$$\frac{1}{x^{79}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(x^{10}\right)^{-7}\times \frac{1}{x^{9}}$$

To raise a power to another power, multiply the exponents.

$$x^{10\left(-7\right)}x^{9\left(-1\right)}$$

Multiply 10 times -7.

$$x^{-70}x^{9\left(-1\right)}$$

Multiply 9 times -1.

$$x^{-70}x^{-9}$$

To multiply powers of the same base, add their exponents.

$$x^{-70-9}$$

Add the exponents -70 and -9.

$$x^{-79}$$

Differentiate w.r.t. x

$$-\frac{79}{x^{80}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 10 and -7 to get -70.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-70}}{x^{9}})$$

Rewrite x^{9} as x^{-70}x^{79}. Cancel out x^{-70} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{79}})$$

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).

$$-\left(x^{79}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{79})$$

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}.

$$-\left(x^{79}\right)^{-2}\times 79x^{79-1}$$

Simplify.

$$-79x^{78}\left(x^{79}\right)^{-2}$$