Solve 4x^2/5*30/x^4*30x^3/60

$$\frac { 4 x ^ { 2 } } { 5 } \times \frac { 30 } { x ^ { 4 } } \times \frac { 30 x ^ { 3 } } { 60 }$$

Evaluate

$$12x$$

Solution Steps

Divide 30x^{3} by 60 to get \frac{1}{2}x^{3}.

$$\frac{4x^{2}}{5}\times \frac{30}{x^{4}}\times \frac{1}{2}x^{3}$$

Multiply \frac{4x^{2}}{5} times \frac{30}{x^{4}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4x^{2}\times 30}{5x^{4}}\times \frac{1}{2}x^{3}$$

Cancel out 5x^{2} in both numerator and denominator.

$$\frac{4\times 6}{x^{2}}\times \frac{1}{2}x^{3}$$

Multiply \frac{4\times 6}{x^{2}} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4\times 6}{x^{2}\times 2}x^{3}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2\times 6}{x^{2}}x^{3}$$

Express \frac{2\times 6}{x^{2}}x^{3} as a single fraction.

$$\frac{2\times 6x^{3}}{x^{2}}$$

Cancel out x^{2} in both numerator and denominator.

$$2\times 6x$$

Multiply 2 and 6 to get 12.

$$12x$$

Differentiate w.r.t. x

$$12$$

Steps Using Definition of a Derivative

Divide 30x^{3} by 60 to get \frac{1}{2}x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x^{2}}{5}\times \frac{30}{x^{4}}\times \frac{1}{2}x^{3})$$

Multiply \frac{4x^{2}}{5} times \frac{30}{x^{4}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x^{2}\times 30}{5x^{4}}\times \frac{1}{2}x^{3})$$

Cancel out 5x^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4\times 6}{x^{2}}\times \frac{1}{2}x^{3})$$

Multiply \frac{4\times 6}{x^{2}} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4\times 6}{x^{2}\times 2}x^{3})$$

Cancel out 2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2\times 6}{x^{2}}x^{3})$$

Express \frac{2\times 6}{x^{2}}x^{3} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2\times 6x^{3}}{x^{2}})$$

Cancel out x^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2\times 6x)$$

Multiply 2 and 6 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(12x)$$

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

$$12x^{1-1}$$

Subtract 1 from 1.

$$12x^{0}$$

For any term t except 0, t^{0}=1.

$$12\times 1$$

For any term t, t\times 1=t and 1t=t.

$$12$$