Solve x/9+x/8+x/7+x/6+frac{x}{5}+frac{x}{4}+frac{x}{3}+frac{x}{2}

$$\frac{ x }{ 9 } + \frac{ x }{ 8 } + \frac{ x }{ 7 } + \frac{ x }{ 6 } + \frac{ x }{ 5 } + \frac{ x }{ 4 } + \frac{ x }{ 3 } + \frac{ x }{ 2 }$$

Evaluate

$$\frac{4609x}{2520}$$

Solution Steps

Combine \frac{x}{9} and \frac{x}{8} to get \frac{17}{72}x.

$$\frac{17}{72}x+\frac{x}{7}+\frac{x}{6}+\frac{x}{5}+\frac{x}{4}+\frac{x}{3}+\frac{x}{2}$$

Combine \frac{17}{72}x and \frac{x}{7} to get \frac{191}{504}x.

$$\frac{191}{504}x+\frac{x}{6}+\frac{x}{5}+\frac{x}{4}+\frac{x}{3}+\frac{x}{2}$$

Combine \frac{191}{504}x and \frac{x}{6} to get \frac{275}{504}x.

$$\frac{275}{504}x+\frac{x}{5}+\frac{x}{4}+\frac{x}{3}+\frac{x}{2}$$

Combine \frac{275}{504}x and \frac{x}{5} to get \frac{1879}{2520}x.

$$\frac{1879}{2520}x+\frac{x}{4}+\frac{x}{3}+\frac{x}{2}$$

Combine \frac{1879}{2520}x and \frac{x}{4} to get \frac{2509}{2520}x.

$$\frac{2509}{2520}x+\frac{x}{3}+\frac{x}{2}$$

Combine \frac{2509}{2520}x and \frac{x}{3} to get \frac{3349}{2520}x.

$$\frac{3349}{2520}x+\frac{x}{2}$$

Combine \frac{3349}{2520}x and \frac{x}{2} to get \frac{4609}{2520}x.

$$\frac{4609}{2520}x$$

Differentiate w.r.t. x

$$\frac{4609}{2520} = 1\frac{2089}{2520} = 1.828968253968254$$

Steps Using Definition of a Derivative

Combine \frac{x}{9} and \frac{x}{8} to get \frac{17}{72}x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{17}{72}x+\frac{x}{7}+\frac{x}{6}+\frac{x}{5}+\frac{x}{4}+\frac{x}{3}+\frac{x}{2})$$

Combine \frac{17}{72}x and \frac{x}{7} to get \frac{191}{504}x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{191}{504}x+\frac{x}{6}+\frac{x}{5}+\frac{x}{4}+\frac{x}{3}+\frac{x}{2})$$

Combine \frac{191}{504}x and \frac{x}{6} to get \frac{275}{504}x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{275}{504}x+\frac{x}{5}+\frac{x}{4}+\frac{x}{3}+\frac{x}{2})$$

Combine \frac{275}{504}x and \frac{x}{5} to get \frac{1879}{2520}x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1879}{2520}x+\frac{x}{4}+\frac{x}{3}+\frac{x}{2})$$

Combine \frac{1879}{2520}x and \frac{x}{4} to get \frac{2509}{2520}x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2509}{2520}x+\frac{x}{3}+\frac{x}{2})$$

Combine \frac{2509}{2520}x and \frac{x}{3} to get \frac{3349}{2520}x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3349}{2520}x+\frac{x}{2})$$

Combine \frac{3349}{2520}x and \frac{x}{2} to get \frac{4609}{2520}x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4609}{2520}x)$$

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

$$\frac{4609}{2520}x^{1-1}$$

Subtract 1 from 1.

$$\frac{4609}{2520}x^{0}$$

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

$$\frac{4609}{2520}\times 1$$

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

$$\frac{4609}{2520}$$