Solve left(x^4right)^2divleft(x^5right)^4*{x}^7

$${ \left( { x }^{ 4 } \right) }^{ 2 } \div { \left( { x }^{ 5 } \right) }^{ 4 } \times { x }^{ 7 }$$

Evaluate

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

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{x^{8}}{\left(x^{5}\right)^{4}}x^{7}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 4 to get 20.

$$\frac{x^{8}}{x^{20}}x^{7}$$

Rewrite x^{20} as x^{8}x^{12}. Cancel out x^{8} in both numerator and denominator.

$$\frac{1}{x^{12}}x^{7}$$

Express \frac{1}{x^{12}}x^{7} as a single fraction.

$$\frac{x^{7}}{x^{12}}$$

Cancel out x^{7} in both numerator and denominator.

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

Differentiate w.r.t. x

$$-\frac{5}{x^{6}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{8}}{\left(x^{5}\right)^{4}}x^{7})$$

To raise a power to another power, multiply the exponents. Multiply 5 and 4 to get 20.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{8}}{x^{20}}x^{7})$$

Rewrite x^{20} as x^{8}x^{12}. Cancel out x^{8} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{12}}x^{7})$$

Express \frac{1}{x^{12}}x^{7} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{7}}{x^{12}})$$

Cancel out x^{7} in both numerator and denominator.

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

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).

$$-\left(x^{5}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{5})$$

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}.

$$-\left(x^{5}\right)^{-2}\times 5x^{5-1}$$

Simplify.

$$-5x^{4}\left(x^{5}\right)^{-2}$$