Solve f(t)=t^3/t^2+20

$$f ( t ) = \frac { t ^ { 3 } } { t ^ { 2 } + 20 }$$

Differentiate w.r.t. t

$$\left(\frac{t}{t^{2}+20}\right)^{2}\left(t^{2}+60\right)$$

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.

$$\frac{\left(t^{2}+20\right)\frac{\mathrm{d}}{\mathrm{d}t}(t^{3})-t^{3}\frac{\mathrm{d}}{\mathrm{d}t}(t^{2}+20)}{\left(t^{2}+20\right)^{2}}$$

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}.

$$\frac{\left(t^{2}+20\right)\times 3t^{3-1}-t^{3}\times 2t^{2-1}}{\left(t^{2}+20\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(t^{2}+20\right)\times 3t^{2}-t^{3}\times 2t^{1}}{\left(t^{2}+20\right)^{2}}$$

Expand using distributive property.

$$\frac{t^{2}\times 3t^{2}+20\times 3t^{2}-t^{3}\times 2t^{1}}{\left(t^{2}+20\right)^{2}}$$

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

$$\frac{3t^{2+2}+20\times 3t^{2}-2t^{3+1}}{\left(t^{2}+20\right)^{2}}$$

Do the arithmetic.

$$\frac{3t^{4}+60t^{2}-2t^{4}}{\left(t^{2}+20\right)^{2}}$$

Combine like terms.

$$\frac{\left(3-2\right)t^{4}+60t^{2}}{\left(t^{2}+20\right)^{2}}$$

Subtract 2 from 3.

$$\frac{t^{4}+60t^{2}}{\left(t^{2}+20\right)^{2}}$$

Factor out t^{2}.

$$\frac{t^{2}\left(t^{2}+60t^{0}\right)}{\left(t^{2}+20\right)^{2}}$$

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

$$\frac{t^{2}\left(t^{2}+60\times 1\right)}{\left(t^{2}+20\right)^{2}}$$

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

$$\frac{t^{2}\left(t^{2}+60\right)}{\left(t^{2}+20\right)^{2}}$$

Evaluate

$$\frac{t^{3}}{t^{2}+20}$$