Solve 1/t^2-3t-1dt

$$\frac { 1 } { t ^ { 2 } - 3 t - 1 } d t$$

Evaluate

$$\frac{dt}{t^{2}-3t-1}$$

Short Solution Steps

Express \frac{1}{t^{2}-3t-1}d as a single fraction.

$$\frac{d}{t^{2}-3t-1}t$$

Express \frac{d}{t^{2}-3t-1}t as a single fraction.

$$\frac{dt}{t^{2}-3t-1}$$

Differentiate w.r.t. t

$$-\frac{d\left(t^{2}+1\right)}{t^{4}-6t^{3}+7t^{2}+6t+1}$$

Steps Using Derivative Rule for Quotient

Express \frac{1}{t^{2}-3t-1}d as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{d}{t^{2}-3t-1}t)$$

Express \frac{d}{t^{2}-3t-1}t as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{dt}{t^{2}-3t-1})$$

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.

$$\frac{\left(t^{2}-3t^{1}-1\right)\frac{\mathrm{d}}{\mathrm{d}t}(dt^{1})-dt^{1}\frac{\mathrm{d}}{\mathrm{d}t}(t^{2}-3t^{1}-1)}{\left(t^{2}-3t^{1}-1\right)^{2}}$$

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}.

$$\frac{\left(t^{2}-3t^{1}-1\right)dt^{1-1}-dt^{1}\left(2t^{2-1}-3t^{1-1}\right)}{\left(t^{2}-3t^{1}-1\right)^{2}}$$

Simplify.

$$\frac{\left(t^{2}-3t^{1}-1\right)dt^{0}-dt^{1}\left(2t^{1}-3t^{0}\right)}{\left(t^{2}-3t^{1}-1\right)^{2}}$$

Multiply t^{2}-3t^{1}-1 times dt^{0}.

$$\frac{t^{2}dt^{0}-3t^{1}dt^{0}-dt^{0}-dt^{1}\left(2t^{1}-3t^{0}\right)}{\left(t^{2}-3t^{1}-1\right)^{2}}$$

Multiply dt^{1} times 2t^{1}-3t^{0}.

$$\frac{t^{2}dt^{0}-3t^{1}dt^{0}-dt^{0}-\left(dt^{1}\times 2t^{1}+dt^{1}\left(-3\right)t^{0}\right)}{\left(t^{2}-3t^{1}-1\right)^{2}}$$

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

$$\frac{dt^{2}-3dt^{1}-dt^{0}-\left(d\times 2t^{1+1}+d\left(-3\right)t^{1}\right)}{\left(t^{2}-3t^{1}-1\right)^{2}}$$

Simplify.

$$\frac{dt^{2}+\left(-3d\right)t^{1}+\left(-d\right)t^{0}-\left(2dt^{2}+\left(-3d\right)t^{1}\right)}{\left(t^{2}-3t^{1}-1\right)^{2}}$$

Combine like terms.

$$\frac{\left(-d\right)t^{2}+\left(-d\right)t^{0}}{\left(t^{2}-3t^{1}-1\right)^{2}}$$

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

$$\frac{\left(-d\right)t^{2}+\left(-d\right)t^{0}}{\left(t^{2}-3t-1\right)^{2}}$$

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

$$\frac{\left(-d\right)t^{2}+\left(-d\right)\times 1}{\left(t^{2}-3t-1\right)^{2}}$$

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

$$\frac{\left(-d\right)t^{2}-d}{\left(t^{2}-3t-1\right)^{2}}$$