Solve 10+3r/4r+9

$$10 + \frac { 3 r } { 4 r + 9 }$$

Evaluate

$$\frac{43r+90}{4r+9}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 10 times \frac{4r+9}{4r+9}.

$$\frac{10\left(4r+9\right)}{4r+9}+\frac{3r}{4r+9}$$

Since \frac{10\left(4r+9\right)}{4r+9} and \frac{3r}{4r+9} have the same denominator, add them by adding their numerators.

$$\frac{10\left(4r+9\right)+3r}{4r+9}$$

Do the multiplications in 10\left(4r+9\right)+3r.

$$\frac{40r+90+3r}{4r+9}$$

Combine like terms in 40r+90+3r.

$$\frac{43r+90}{4r+9}$$

Differentiate w.r.t. r

$$\frac{27}{\left(4r+9\right)^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Multiply 10 times \frac{4r+9}{4r+9}.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{10\left(4r+9\right)}{4r+9}+\frac{3r}{4r+9})$$

Since \frac{10\left(4r+9\right)}{4r+9} and \frac{3r}{4r+9} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{10\left(4r+9\right)+3r}{4r+9})$$

Do the multiplications in 10\left(4r+9\right)+3r.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{40r+90+3r}{4r+9})$$

Combine like terms in 40r+90+3r.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{43r+90}{4r+9})$$

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.

$$\frac{\left(4r^{1}+9\right)\frac{\mathrm{d}}{\mathrm{d}r}(43r^{1}+90)-\left(43r^{1}+90\right)\frac{\mathrm{d}}{\mathrm{d}r}(4r^{1}+9)}{\left(4r^{1}+9\right)^{2}}$$

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}.

$$\frac{\left(4r^{1}+9\right)\times 43r^{1-1}-\left(43r^{1}+90\right)\times 4r^{1-1}}{\left(4r^{1}+9\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(4r^{1}+9\right)\times 43r^{0}-\left(43r^{1}+90\right)\times 4r^{0}}{\left(4r^{1}+9\right)^{2}}$$

Expand using distributive property.

$$\frac{4r^{1}\times 43r^{0}+9\times 43r^{0}-\left(43r^{1}\times 4r^{0}+90\times 4r^{0}\right)}{\left(4r^{1}+9\right)^{2}}$$

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

$$\frac{4\times 43r^{1}+9\times 43r^{0}-\left(43\times 4r^{1}+90\times 4r^{0}\right)}{\left(4r^{1}+9\right)^{2}}$$

Do the arithmetic.

$$\frac{172r^{1}+387r^{0}-\left(172r^{1}+360r^{0}\right)}{\left(4r^{1}+9\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{172r^{1}+387r^{0}-172r^{1}-360r^{0}}{\left(4r^{1}+9\right)^{2}}$$

Combine like terms.

$$\frac{\left(172-172\right)r^{1}+\left(387-360\right)r^{0}}{\left(4r^{1}+9\right)^{2}}$$

Subtract 172 from 172 and 360 from 387.

$$\frac{27r^{0}}{\left(4r^{1}+9\right)^{2}}$$

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

$$\frac{27r^{0}}{\left(4r+9\right)^{2}}$$

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

$$\frac{27\times 1}{\left(4r+9\right)^{2}}$$

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

$$\frac{27}{\left(4r+9\right)^{2}}$$