Solve 4/x*x^3/2*x/10

$$\frac { 4 } { x } \times \frac { x ^ { 3 } } { 2 } \times \frac { x } { 10 }$$

Evaluate

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

Solution Steps

Multiply \frac{4}{x} times \frac{x^{3}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4x^{3}}{x\times 2}\times \frac{x}{10}$$

Cancel out 2x in both numerator and denominator.

$$2x^{2}\times \frac{x}{10}$$

Express 2x^{2}\times \frac{x}{10} as a single fraction.

$$\frac{2x^{2}x}{10}$$

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

$$\frac{2x^{3}}{10}$$

Divide 2x^{3} by 10 to get \frac{1}{5}x^{3}.

$$\frac{1}{5}x^{3}$$

Differentiate w.r.t. x

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

Steps Using Definition of a Derivative

Multiply \frac{4}{x} times \frac{x^{3}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x^{3}}{x\times 2}\times \frac{x}{10})$$

Cancel out 2x in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{2}\times \frac{x}{10})$$

Express 2x^{2}\times \frac{x}{10} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{2}x}{10})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{3}}{10})$$

Divide 2x^{3} by 10 to get \frac{1}{5}x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{5}x^{3})$$

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

$$3\times \frac{1}{5}x^{3-1}$$

Multiply 3 times \frac{1}{5}.

$$\frac{3}{5}x^{3-1}$$

Subtract 1 from 3.

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