Solve x^-11*x^4/x^-3

$$\frac{ { x }^{ -11 } \times { x }^{ 4 } }{ { x }^{ -3 } }$$

Evaluate

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

Solution Steps

To multiply powers of the same base, add their exponents. Add -11 and 4 to get -7.

$$\frac{x^{-7}}{x^{-3}}$$

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

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

Differentiate w.r.t. x

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

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add -11 and 4 to get -7.

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

Rewrite x^{-3} as x^{-7}x^{4}. Cancel out x^{-7} 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.

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