Solve [3/4*0001t^4-1/3*001t^3-1/203t^2+4t]_0^12

$$[ \frac { 3 } { 4 } \cdot 0001 t ^ { 4 } - \frac { 1 } { 3 } \cdot 001 t ^ { 3 } - \frac { 1 } { 2 } 03 t ^ { 2 } + 4 t ] _ { 0 } ^ { 12 }$$

Evaluate

$$4t$$

Solution Steps

Multiply \frac{3}{4} and 0 to get 0.

$$0\times 0\times 0\times 1t^{4}-\frac{1}{3}\times 0\times 0\times 1t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t$$

Multiply 0 and 0 to get 0.

$$0\times 0\times 1t^{4}-\frac{1}{3}\times 0\times 0\times 1t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t$$

Multiply 0 and 0 to get 0.

$$0\times 1t^{4}-\frac{1}{3}\times 0\times 0\times 1t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t$$

Multiply 0 and 1 to get 0.

$$0t^{4}-\frac{1}{3}\times 0\times 0\times 1t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t$$

Anything times zero gives zero.

$$0-\frac{1}{3}\times 0\times 0\times 1t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t$$

Multiply \frac{1}{3} and 0 to get 0.

$$0-0\times 0\times 1t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t$$

Multiply 0 and 0 to get 0.

$$0-0\times 1t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t$$

Multiply 0 and 1 to get 0.

$$0-0t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t$$

Anything times zero gives zero.

$$0-0-\frac{1}{2}\times 0\times 3t^{2}+4t$$

Subtracting 0 from itself leaves 0.

$$0-\frac{1}{2}\times 0\times 3t^{2}+4t$$

Multiply \frac{1}{2} and 0 to get 0.

$$0-0\times 3t^{2}+4t$$

Multiply 0 and 3 to get 0.

$$0-0t^{2}+4t$$

Anything times zero gives zero.

$$0-0+4t$$

Subtracting 0 from itself leaves 0.

$$0+4t$$

Anything plus zero gives itself.

$$4t$$

Differentiate w.r.t. t

$$4$$

Steps Using Definition of a Derivative

Multiply \frac{3}{4} and 0 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}t}(0\times 0\times 0\times 1t^{4}-\frac{1}{3}\times 0\times 0\times 1t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t)$$

Multiply 0 and 0 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}t}(0\times 0\times 1t^{4}-\frac{1}{3}\times 0\times 0\times 1t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t)$$

Multiply 0 and 0 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}t}(0\times 1t^{4}-\frac{1}{3}\times 0\times 0\times 1t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t)$$

Multiply 0 and 1 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}t}(0t^{4}-\frac{1}{3}\times 0\times 0\times 1t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t)$$

Anything times zero gives zero.

$$\frac{\mathrm{d}}{\mathrm{d}t}(0-\frac{1}{3}\times 0\times 0\times 1t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t)$$

Multiply \frac{1}{3} and 0 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}t}(0-0\times 0\times 1t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t)$$

Multiply 0 and 0 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}t}(0-0\times 1t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t)$$

Multiply 0 and 1 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}t}(0-0t^{3}-\frac{1}{2}\times 0\times 3t^{2}+4t)$$

Anything times zero gives zero.

$$\frac{\mathrm{d}}{\mathrm{d}t}(0-0-\frac{1}{2}\times 0\times 3t^{2}+4t)$$

Subtracting 0 from itself leaves 0.

$$\frac{\mathrm{d}}{\mathrm{d}t}(0-\frac{1}{2}\times 0\times 3t^{2}+4t)$$

Multiply \frac{1}{2} and 0 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}t}(0-0\times 3t^{2}+4t)$$

Multiply 0 and 3 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}t}(0-0t^{2}+4t)$$

Anything times zero gives zero.

$$\frac{\mathrm{d}}{\mathrm{d}t}(0-0+4t)$$

Subtracting 0 from itself leaves 0.

$$\frac{\mathrm{d}}{\mathrm{d}t}(0+4t)$$

Anything plus zero gives itself.

$$\frac{\mathrm{d}}{\mathrm{d}t}(4t)$$

The derivative of ax^{n} is nax^{n-1}.

$$4t^{1-1}$$

Subtract 1 from 1.

$$4t^{0}$$

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

$$4\times 1$$

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

$$4$$