Solve frac{5x^frac{3{5}}*5x^frac{12{5}}}{(5x^frac{-3{2}})^2}

$$\frac { 5 x ^ { \frac { 3 } { 5 } } \cdot 5 x ^ { \frac { 12 } { 5 } } } { ( 5 x ^ { \frac { - 3 } { 2 } } ) ^ { 2 } }$$

Evaluate

$$x^{6}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{3}{5} and \frac{12}{5} to get 3.

$$\frac{5x^{3}\times 5}{\left(5x^{\frac{-3}{2}}\right)^{2}}$$

Multiply 5 and 5 to get 25.

$$\frac{25x^{3}}{\left(5x^{\frac{-3}{2}}\right)^{2}}$$

Fraction \frac{-3}{2} can be rewritten as -\frac{3}{2} by extracting the negative sign.

$$\frac{25x^{3}}{\left(5x^{-\frac{3}{2}}\right)^{2}}$$

Expand \left(5x^{-\frac{3}{2}}\right)^{2}.

$$\frac{25x^{3}}{5^{2}\left(x^{-\frac{3}{2}}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply -\frac{3}{2} and 2 to get -3.

$$\frac{25x^{3}}{5^{2}x^{-3}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{25x^{3}}{25x^{-3}}$$

Cancel out 25 in both numerator and denominator.

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

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

$$x^{6}$$

Differentiate w.r.t. x

$$6x^{5}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{3}{5} and \frac{12}{5} to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x^{3}\times 5}{\left(5x^{\frac{-3}{2}}\right)^{2}})$$

Multiply 5 and 5 to get 25.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{25x^{3}}{\left(5x^{\frac{-3}{2}}\right)^{2}})$$

Fraction \frac{-3}{2} can be rewritten as -\frac{3}{2} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{25x^{3}}{\left(5x^{-\frac{3}{2}}\right)^{2}})$$

Expand \left(5x^{-\frac{3}{2}}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{25x^{3}}{5^{2}\left(x^{-\frac{3}{2}}\right)^{2}})$$

To raise a power to another power, multiply the exponents. Multiply -\frac{3}{2} and 2 to get -3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{25x^{3}}{5^{2}x^{-3}})$$

Calculate 5 to the power of 2 and get 25.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{25x^{3}}{25x^{-3}})$$

Cancel out 25 in both numerator and denominator.

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

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6})$$

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

$$6x^{6-1}$$

Subtract 1 from 6.

$$6x^{5}$$