Solve 7(25x^4)^-frac{1{2}}

$$7 ( 25 x ^ { 4 } ) ^ { - \frac { 1 } { 2 } }$$

Evaluate

$$\frac{7}{5x^{2}}$$

Solution Steps

Expand \left(25x^{4}\right)^{-\frac{1}{2}}.

$$7\times 25^{-\frac{1}{2}}\left(x^{4}\right)^{-\frac{1}{2}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -\frac{1}{2} to get -2.

$$7\times 25^{-\frac{1}{2}}x^{-2}$$

Calculate 25 to the power of -\frac{1}{2} and get \frac{1}{5}.

$$7\times \frac{1}{5}x^{-2}$$

Multiply 7 and \frac{1}{5} to get \frac{7}{5}.

$$\frac{7}{5}x^{-2}$$

Differentiate w.r.t. x

$$-\frac{14}{5x^{3}}$$

Steps Using Definition of a Derivative

Expand \left(25x^{4}\right)^{-\frac{1}{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(7\times 25^{-\frac{1}{2}}\left(x^{4}\right)^{-\frac{1}{2}})$$

To raise a power to another power, multiply the exponents. Multiply 4 and -\frac{1}{2} to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(7\times 25^{-\frac{1}{2}}x^{-2})$$

Calculate 25 to the power of -\frac{1}{2} and get \frac{1}{5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(7\times \frac{1}{5}x^{-2})$$

Multiply 7 and \frac{1}{5} to get \frac{7}{5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7}{5}x^{-2})$$

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

$$-2\times \frac{7}{5}x^{-2-1}$$

Multiply -2 times \frac{7}{5}.

$$-\frac{14}{5}x^{-2-1}$$

Subtract 1 from -2.

$$-\frac{14}{5}x^{-3}$$