Solve d/dxleft(1/frac{1{x}+1/7+1/8}right)

$$\frac{d}{d x } \left( \frac{ 1 }{ \frac{ 1 }{ x } + \frac{ 1 }{ 7 } + \frac{ 1 }{ 8 } } \right)$$

Evaluate

$$\frac{3136}{\left(15x+56\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Least common multiple of 7 and 8 is 56. Convert \frac{1}{7} and \frac{1}{8} to fractions with denominator 56.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{\frac{1}{x}+\frac{8}{56}+\frac{7}{56}})$$

Since \frac{8}{56} and \frac{7}{56} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{\frac{1}{x}+\frac{8+7}{56}})$$

Add 8 and 7 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{\frac{1}{x}+\frac{15}{56}})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and 56 is 56x. Multiply \frac{1}{x} times \frac{56}{56}. Multiply \frac{15}{56} times \frac{x}{x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{\frac{56}{56x}+\frac{15x}{56x}})$$

Since \frac{56}{56x} and \frac{15x}{56x} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{\frac{56+15x}{56x}})$$

Divide 1 by \frac{56+15x}{56x} by multiplying 1 by the reciprocal of \frac{56+15x}{56x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{56x}{56+15x})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{\left(15x^{1}+56\right)\frac{\mathrm{d}}{\mathrm{d}x}(56x^{1})-56x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(15x^{1}+56)}{\left(15x^{1}+56\right)^{2}}$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\frac{\left(15x^{1}+56\right)\times 56x^{1-1}-56x^{1}\times 15x^{1-1}}{\left(15x^{1}+56\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(15x^{1}+56\right)\times 56x^{0}-56x^{1}\times 15x^{0}}{\left(15x^{1}+56\right)^{2}}$$

Expand using distributive property.

$$\frac{15x^{1}\times 56x^{0}+56\times 56x^{0}-56x^{1}\times 15x^{0}}{\left(15x^{1}+56\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{15\times 56x^{1}+56\times 56x^{0}-56\times 15x^{1}}{\left(15x^{1}+56\right)^{2}}$$

Do the arithmetic.

$$\frac{840x^{1}+3136x^{0}-840x^{1}}{\left(15x^{1}+56\right)^{2}}$$

Combine like terms.

$$\frac{\left(840-840\right)x^{1}+3136x^{0}}{\left(15x^{1}+56\right)^{2}}$$

Subtract 840 from 840.

$$\frac{3136x^{0}}{\left(15x^{1}+56\right)^{2}}$$

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

$$\frac{3136x^{0}}{\left(15x+56\right)^{2}}$$

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

$$\frac{3136\times 1}{\left(15x+56\right)^{2}}$$

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

$$\frac{3136}{\left(15x+56\right)^{2}}$$

Differentiate w.r.t. x

$$-\frac{94080}{\left(15x+56\right)^{3}}$$