Solve v(t)=4t^2/4+t^3

$$v ( t ) = \frac { 4 t ^ { 2 } } { 4 + t ^ { 3 } }$$

Differentiate w.r.t. t

$$\frac{4t\left(8-t^{3}\right)}{\left(t^{3}+4\right)^{2}}$$

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.

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

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}.

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

Do the arithmetic.

$$\frac{\left(t^{3}+4\right)\times 8t^{1}-4t^{2}\times 3t^{2}}{\left(t^{3}+4\right)^{2}}$$

Expand using distributive property.

$$\frac{t^{3}\times 8t^{1}+4\times 8t^{1}-4t^{2}\times 3t^{2}}{\left(t^{3}+4\right)^{2}}$$

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

$$\frac{8t^{3+1}+4\times 8t^{1}-4\times 3t^{2+2}}{\left(t^{3}+4\right)^{2}}$$

Do the arithmetic.

$$\frac{8t^{4}+32t^{1}-12t^{4}}{\left(t^{3}+4\right)^{2}}$$

Combine like terms.

$$\frac{\left(8-12\right)t^{4}+32t^{1}}{\left(t^{3}+4\right)^{2}}$$

Subtract 12 from 8.

$$\frac{-4t^{4}+32t^{1}}{\left(t^{3}+4\right)^{2}}$$

Factor out 4t.

$$\frac{4t\left(-t^{3}+8t^{0}\right)}{\left(t^{3}+4\right)^{2}}$$

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

$$\frac{4t\left(-t^{3}+8\times 1\right)}{\left(t^{3}+4\right)^{2}}$$

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

$$\frac{4t\left(-t^{3}+8\right)}{\left(t^{3}+4\right)^{2}}$$

Evaluate

$$\frac{4t^{2}}{t^{3}+4}$$