Solve 2r/r+10+5

$$\frac { 2 r } { r + 10 } + 5$$

Evaluate

$$\frac{7r+50}{r+10}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{r+10}{r+10}.

$$\frac{2r}{r+10}+\frac{5\left(r+10\right)}{r+10}$$

Since \frac{2r}{r+10} and \frac{5\left(r+10\right)}{r+10} have the same denominator, add them by adding their numerators.

$$\frac{2r+5\left(r+10\right)}{r+10}$$

Do the multiplications in 2r+5\left(r+10\right).

$$\frac{2r+5r+50}{r+10}$$

Combine like terms in 2r+5r+50.

$$\frac{7r+50}{r+10}$$

Differentiate w.r.t. r

$$\frac{20}{\left(r+10\right)^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{r+10}{r+10}.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{2r}{r+10}+\frac{5\left(r+10\right)}{r+10})$$

Since \frac{2r}{r+10} and \frac{5\left(r+10\right)}{r+10} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{2r+5\left(r+10\right)}{r+10})$$

Do the multiplications in 2r+5\left(r+10\right).

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{2r+5r+50}{r+10})$$

Combine like terms in 2r+5r+50.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{7r+50}{r+10})$$

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.

$$\frac{\left(r^{1}+10\right)\frac{\mathrm{d}}{\mathrm{d}r}(7r^{1}+50)-\left(7r^{1}+50\right)\frac{\mathrm{d}}{\mathrm{d}r}(r^{1}+10)}{\left(r^{1}+10\right)^{2}}$$

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}.

$$\frac{\left(r^{1}+10\right)\times 7r^{1-1}-\left(7r^{1}+50\right)r^{1-1}}{\left(r^{1}+10\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(r^{1}+10\right)\times 7r^{0}-\left(7r^{1}+50\right)r^{0}}{\left(r^{1}+10\right)^{2}}$$

Expand using distributive property.

$$\frac{r^{1}\times 7r^{0}+10\times 7r^{0}-\left(7r^{1}r^{0}+50r^{0}\right)}{\left(r^{1}+10\right)^{2}}$$

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

$$\frac{7r^{1}+10\times 7r^{0}-\left(7r^{1}+50r^{0}\right)}{\left(r^{1}+10\right)^{2}}$$

Do the arithmetic.

$$\frac{7r^{1}+70r^{0}-\left(7r^{1}+50r^{0}\right)}{\left(r^{1}+10\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{7r^{1}+70r^{0}-7r^{1}-50r^{0}}{\left(r^{1}+10\right)^{2}}$$

Combine like terms.

$$\frac{\left(7-7\right)r^{1}+\left(70-50\right)r^{0}}{\left(r^{1}+10\right)^{2}}$$

Subtract 7 from 7 and 50 from 70.

$$\frac{20r^{0}}{\left(r^{1}+10\right)^{2}}$$

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

$$\frac{20r^{0}}{\left(r+10\right)^{2}}$$

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

$$\frac{20\times 1}{\left(r+10\right)^{2}}$$

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

$$\frac{20}{\left(r+10\right)^{2}}$$