Solve frac{{left(sqrt{x^4}right)}^{2}*{left(sqrt{x^8}right)}^{2}{left(sqrt{x^14}right)}^{2}}{{left(sqrt{x^4}right)}^{2}*{left(sqrt{{x}^8}right)}^2*{left(sqrt{{x}^2}right)}^2}

$$\frac{ { \left( \sqrt{ { x }^{ 4 } } \right) }^{ 2 } \times { \left( \sqrt{ { x }^{ 8 } } \right) }^{ 2 } { \left( \sqrt{ { x }^{ 14 } } \right) }^{ 2 } }{ { \left( \sqrt{ { x }^{ 4 } } \right) }^{ 2 } \times { \left( \sqrt{ { x }^{ 8 } } \right) }^{ 2 } \times { \left( \sqrt{ { x }^{ 2 } } \right) }^{ 2 } }$$

Evaluate

$$x^{12}$$

Solution Steps

Cancel out \left(\sqrt{x^{4}}\right)^{2}\left(\sqrt{x^{8}}\right)^{2} in both numerator and denominator.

$$\frac{\left(\sqrt{x^{14}}\right)^{2}}{\left(\sqrt{x^{2}}\right)^{2}}$$

Calculate \sqrt{x^{14}} to the power of 2 and get x^{14}.

$$\frac{x^{14}}{\left(\sqrt{x^{2}}\right)^{2}}$$

Calculate \sqrt{x^{2}} to the power of 2 and get x^{2}.

$$\frac{x^{14}}{x^{2}}$$

Cancel out x^{2} in both numerator and denominator.

$$x^{12}$$

Differentiate w.r.t. x

$$12x^{11}$$

Steps Using Definition of a Derivative

Cancel out \left(\sqrt{x^{4}}\right)^{2}\left(\sqrt{x^{8}}\right)^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(\sqrt{x^{14}}\right)^{2}}{\left(\sqrt{x^{2}}\right)^{2}})$$

Calculate \sqrt{x^{14}} to the power of 2 and get x^{14}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{14}}{\left(\sqrt{x^{2}}\right)^{2}})$$

Calculate \sqrt{x^{2}} to the power of 2 and get x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{14}}{x^{2}})$$

Cancel out x^{2} in both numerator and denominator.

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

The derivative of ax^{n} is nax^{n-1}.

$$12x^{12-1}$$

Subtract 1 from 12.

$$12x^{11}$$