Solve 20t/5t-1

$$\frac { 20 t } { 5 t - 1 }$$

Differentiate w.r.t. t

$$-\frac{20}{\left(5t-1\right)^{2}}$$

Steps Using Derivative Rule for Quotient

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.

$$\frac{\left(5t^{1}-1\right)\frac{\mathrm{d}}{\mathrm{d}t}(20t^{1})-20t^{1}\frac{\mathrm{d}}{\mathrm{d}t}(5t^{1}-1)}{\left(5t^{1}-1\right)^{2}}$$

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}.

$$\frac{\left(5t^{1}-1\right)\times 20t^{1-1}-20t^{1}\times 5t^{1-1}}{\left(5t^{1}-1\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(5t^{1}-1\right)\times 20t^{0}-20t^{1}\times 5t^{0}}{\left(5t^{1}-1\right)^{2}}$$

Expand using distributive property.

$$\frac{5t^{1}\times 20t^{0}-20t^{0}-20t^{1}\times 5t^{0}}{\left(5t^{1}-1\right)^{2}}$$

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

$$\frac{5\times 20t^{1}-20t^{0}-20\times 5t^{1}}{\left(5t^{1}-1\right)^{2}}$$

Do the arithmetic.

$$\frac{100t^{1}-20t^{0}-100t^{1}}{\left(5t^{1}-1\right)^{2}}$$

Combine like terms.

$$\frac{\left(100-100\right)t^{1}-20t^{0}}{\left(5t^{1}-1\right)^{2}}$$

Subtract 100 from 100.

$$\frac{-20t^{0}}{\left(5t^{1}-1\right)^{2}}$$

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

$$\frac{-20t^{0}}{\left(5t-1\right)^{2}}$$

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

$$\frac{-20}{\left(5t-1\right)^{2}}$$

Evaluate

$$\frac{20t}{5t-1}$$