Solve [1-(1305/1300+185x+1955)]*100%

$$[ 1 - ( \frac { 1305 } { 1300 + 185 x + 1955 } ) ] \times 100 \%$$

Evaluate

$$\frac{37x+390}{37x+651}$$

Short Solution Steps

Divide 100 by 100 to get 1.

$$\left(1-\frac{1305}{1300+185x+1955}\right)\times 1$$

Add 1300 and 1955 to get 3255.

$$\left(1-\frac{1305}{3255+185x}\right)\times 1$$

Factor 3255+185x.

$$\left(1-\frac{1305}{5\left(37x+651\right)}\right)\times 1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{5\left(37x+651\right)}{5\left(37x+651\right)}.

$$\left(\frac{5\left(37x+651\right)}{5\left(37x+651\right)}-\frac{1305}{5\left(37x+651\right)}\right)\times 1$$

Since \frac{5\left(37x+651\right)}{5\left(37x+651\right)} and \frac{1305}{5\left(37x+651\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\left(37x+651\right)-1305}{5\left(37x+651\right)}\times 1$$

Do the multiplications in 5\left(37x+651\right)-1305.

$$\frac{185x+3255-1305}{5\left(37x+651\right)}\times 1$$

Combine like terms in 185x+3255-1305.

$$\frac{185x+1950}{5\left(37x+651\right)}\times 1$$

Factor the expressions that are not already factored in \frac{185x+1950}{5\left(37x+651\right)}.

$$\frac{5\left(37x+390\right)}{5\left(37x+651\right)}\times 1$$

Cancel out 5 in both numerator and denominator.

$$\frac{37x+390}{37x+651}\times 1$$

Express \frac{37x+390}{37x+651}\times 1 as a single fraction.

$$\frac{37x+390}{37x+651}$$

Differentiate w.r.t. x

$$\frac{9657}{\left(37x+651\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Divide 100 by 100 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(1-\frac{1305}{1300+185x+1955}\right)\times 1)$$

Add 1300 and 1955 to get 3255.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(1-\frac{1305}{3255+185x}\right)\times 1)$$

Factor 3255+185x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(1-\frac{1305}{5\left(37x+651\right)}\right)\times 1)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{5\left(37x+651\right)}{5\left(37x+651\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{5\left(37x+651\right)}{5\left(37x+651\right)}-\frac{1305}{5\left(37x+651\right)}\right)\times 1)$$

Since \frac{5\left(37x+651\right)}{5\left(37x+651\right)} and \frac{1305}{5\left(37x+651\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5\left(37x+651\right)-1305}{5\left(37x+651\right)}\times 1)$$

Do the multiplications in 5\left(37x+651\right)-1305.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{185x+3255-1305}{5\left(37x+651\right)}\times 1)$$

Combine like terms in 185x+3255-1305.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{185x+1950}{5\left(37x+651\right)}\times 1)$$

Factor the expressions that are not already factored in \frac{185x+1950}{5\left(37x+651\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5\left(37x+390\right)}{5\left(37x+651\right)}\times 1)$$

Cancel out 5 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{37x+390}{37x+651}\times 1)$$

Express \frac{37x+390}{37x+651}\times 1 as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{37x+390}{37x+651})$$

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.

$$\frac{\left(37x^{1}+651\right)\frac{\mathrm{d}}{\mathrm{d}x}(37x^{1}+390)-\left(37x^{1}+390\right)\frac{\mathrm{d}}{\mathrm{d}x}(37x^{1}+651)}{\left(37x^{1}+651\right)^{2}}$$

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}.

$$\frac{\left(37x^{1}+651\right)\times 37x^{1-1}-\left(37x^{1}+390\right)\times 37x^{1-1}}{\left(37x^{1}+651\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(37x^{1}+651\right)\times 37x^{0}-\left(37x^{1}+390\right)\times 37x^{0}}{\left(37x^{1}+651\right)^{2}}$$

Expand using distributive property.

$$\frac{37x^{1}\times 37x^{0}+651\times 37x^{0}-\left(37x^{1}\times 37x^{0}+390\times 37x^{0}\right)}{\left(37x^{1}+651\right)^{2}}$$

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

$$\frac{37\times 37x^{1}+651\times 37x^{0}-\left(37\times 37x^{1}+390\times 37x^{0}\right)}{\left(37x^{1}+651\right)^{2}}$$

Do the arithmetic.

$$\frac{1369x^{1}+24087x^{0}-\left(1369x^{1}+14430x^{0}\right)}{\left(37x^{1}+651\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{1369x^{1}+24087x^{0}-1369x^{1}-14430x^{0}}{\left(37x^{1}+651\right)^{2}}$$

Combine like terms.

$$\frac{\left(1369-1369\right)x^{1}+\left(24087-14430\right)x^{0}}{\left(37x^{1}+651\right)^{2}}$$

Subtract 1369 from 1369 and 14430 from 24087.

$$\frac{9657x^{0}}{\left(37x^{1}+651\right)^{2}}$$

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

$$\frac{9657x^{0}}{\left(37x+651\right)^{2}}$$

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

$$\frac{9657\times 1}{\left(37x+651\right)^{2}}$$

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

$$\frac{9657}{\left(37x+651\right)^{2}}$$