Solve (x^3)^-7

$$( x ^ { 3 } ) ^ { - 7 }$$

Evaluate

$$\frac{1}{x^{21}}$$

Steps for Raising Power to Another Power

Use the rules of exponents to simplify the expression.

$$\left(x^{3}\right)^{-7}$$

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

$$x^{3\left(-7\right)}$$

Multiply 3 times -7.

$$\frac{1}{x^{21}}$$

Differentiate w.r.t. x

$$-\frac{21}{x^{22}}$$

Steps Using Chain Rule

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).

$$-7\left(x^{3}\right)^{-7-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{3})$$

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}.

$$-7\left(x^{3}\right)^{-8}\times 3x^{3-1}$$

Simplify.

$$-21x^{2}\left(x^{3}\right)^{-8}$$