Solve x4/5x-2/5x3/5

$$x \frac{ 4 }{ 5 } x \frac{ -2 }{ 5 } x \frac{ 3 }{ 5 }$$

Evaluate

$$-\frac{24x^{3}}{125}$$

Solution Steps

Multiply x and x to get x^{2}.

$$x^{2}\times \frac{4}{5}\times \frac{-2}{5}x\times \frac{3}{5}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{3}\times \frac{4}{5}\times \frac{-2}{5}\times \frac{3}{5}$$

Fraction \frac{-2}{5} can be rewritten as -\frac{2}{5} by extracting the negative sign.

$$x^{3}\times \frac{4}{5}\left(-\frac{2}{5}\right)\times \frac{3}{5}$$

Multiply \frac{4}{5} times -\frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$x^{3}\times \frac{4\left(-2\right)}{5\times 5}\times \frac{3}{5}$$

Do the multiplications in the fraction \frac{4\left(-2\right)}{5\times 5}.

$$x^{3}\times \frac{-8}{25}\times \frac{3}{5}$$

Fraction \frac{-8}{25} can be rewritten as -\frac{8}{25} by extracting the negative sign.

$$x^{3}\left(-\frac{8}{25}\right)\times \frac{3}{5}$$

Multiply -\frac{8}{25} times \frac{3}{5} by multiplying numerator times numerator and denominator times denominator.

$$x^{3}\times \frac{-8\times 3}{25\times 5}$$

Do the multiplications in the fraction \frac{-8\times 3}{25\times 5}.

$$x^{3}\times \frac{-24}{125}$$

Fraction \frac{-24}{125} can be rewritten as -\frac{24}{125} by extracting the negative sign.

$$x^{3}\left(-\frac{24}{125}\right)$$

Differentiate w.r.t. x

$$-\frac{72x^{2}}{125}$$

Steps Using Definition of a Derivative

Multiply x and x to get x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}\times \frac{4}{5}\times \frac{-2}{5}x\times \frac{3}{5})$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}\times \frac{4}{5}\times \frac{-2}{5}\times \frac{3}{5})$$

Fraction \frac{-2}{5} can be rewritten as -\frac{2}{5} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}\times \frac{4}{5}\left(-\frac{2}{5}\right)\times \frac{3}{5})$$

Multiply \frac{4}{5} times -\frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}\times \frac{4\left(-2\right)}{5\times 5}\times \frac{3}{5})$$

Do the multiplications in the fraction \frac{4\left(-2\right)}{5\times 5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}\times \frac{-8}{25}\times \frac{3}{5})$$

Fraction \frac{-8}{25} can be rewritten as -\frac{8}{25} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}\left(-\frac{8}{25}\right)\times \frac{3}{5})$$

Multiply -\frac{8}{25} times \frac{3}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}\times \frac{-8\times 3}{25\times 5})$$

Do the multiplications in the fraction \frac{-8\times 3}{25\times 5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}\times \frac{-24}{125})$$

Fraction \frac{-24}{125} can be rewritten as -\frac{24}{125} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}\left(-\frac{24}{125}\right))$$

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

$$3\left(-\frac{24}{125}\right)x^{3-1}$$

Multiply 3 times -\frac{24}{125}.

$$-\frac{72}{125}x^{3-1}$$

Subtract 1 from 3.

$$-\frac{72}{125}x^{2}$$