Solve 530t/200+t

$$5 \frac{ 30t }{ 200+t }$$

Evaluate

$$\frac{150t}{t+200}$$

Solution Steps

Express 5\times \frac{30t}{200+t} as a single fraction.

$$\frac{5\times 30t}{200+t}$$

Multiply 5 and 30 to get 150.

$$\frac{150t}{200+t}$$

Differentiate w.r.t. t

$$\frac{30000}{\left(t+200\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Express 5\times \frac{30t}{200+t} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{5\times 30t}{200+t})$$

Multiply 5 and 30 to get 150.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{150t}{200+t})$$

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.

$$\frac{\left(t^{1}+200\right)\frac{\mathrm{d}}{\mathrm{d}t}(150t^{1})-150t^{1}\frac{\mathrm{d}}{\mathrm{d}t}(t^{1}+200)}{\left(t^{1}+200\right)^{2}}$$

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}.

$$\frac{\left(t^{1}+200\right)\times 150t^{1-1}-150t^{1}t^{1-1}}{\left(t^{1}+200\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(t^{1}+200\right)\times 150t^{0}-150t^{1}t^{0}}{\left(t^{1}+200\right)^{2}}$$

Expand using distributive property.

$$\frac{t^{1}\times 150t^{0}+200\times 150t^{0}-150t^{1}t^{0}}{\left(t^{1}+200\right)^{2}}$$

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

$$\frac{150t^{1}+200\times 150t^{0}-150t^{1}}{\left(t^{1}+200\right)^{2}}$$

Do the arithmetic.

$$\frac{150t^{1}+30000t^{0}-150t^{1}}{\left(t^{1}+200\right)^{2}}$$

Combine like terms.

$$\frac{\left(150-150\right)t^{1}+30000t^{0}}{\left(t^{1}+200\right)^{2}}$$

Subtract 150 from 150.

$$\frac{30000t^{0}}{\left(t^{1}+200\right)^{2}}$$

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

$$\frac{30000t^{0}}{\left(t+200\right)^{2}}$$

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

$$\frac{30000\times 1}{\left(t+200\right)^{2}}$$

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

$$\frac{30000}{\left(t+200\right)^{2}}$$