Solve x^-2left(5^-1right)^2/5*5^-2

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

Evaluate

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

Solution Steps

To raise a power to another power, multiply the exponents. Multiply -1 and 2 to get -2.

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

To multiply powers of the same base, add their exponents. Add 1 and -2 to get -1.

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

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

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

Calculate 5 to the power of 1 and get 5.

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

Differentiate w.r.t. x

$$-\frac{2}{5x^{3}}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply -1 and 2 to get -2.

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

To multiply powers of the same base, add their exponents. Add 1 and -2 to get -1.

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

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

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

Calculate 5 to the power of 1 and get 5.

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

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

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

Multiply -2 times \frac{1}{5}.

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

Subtract 1 from -2.

$$-\frac{2}{5}x^{-3}$$