Solve f(x)=frac{x^{7}}{42}-x^6/30-x^5/20+x^4/12+3x+2Rightarrowf^primeprime(x)=?

$$f ( x ) = \frac { x ^ { 7 } } { 42 } - \frac { x ^ { 6 } } { 30 } - \frac { x ^ { 5 } } { 20 } + \frac { x ^ { 4 } } { 12 } + 3 x + 2 \Rightarrow f ^ { \prime \prime } ( x ) = ?$$

Evaluate

$$\frac{x^{7}}{42}-\frac{x^{6}}{30}-\frac{x^{5}}{20}+\frac{x^{4}}{12}+3x+2$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 42 and 30 is 210. Multiply \frac{x^{7}}{42} times \frac{5}{5}. Multiply \frac{x^{6}}{30} times \frac{7}{7}.

$$\frac{5x^{7}}{210}-\frac{7x^{6}}{210}-\frac{x^{5}}{20}+\frac{x^{4}}{12}+3x+2$$

Since \frac{5x^{7}}{210} and \frac{7x^{6}}{210} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5x^{7}-7x^{6}}{210}-\frac{x^{5}}{20}+\frac{x^{4}}{12}+3x+2$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 210 and 20 is 420. Multiply \frac{5x^{7}-7x^{6}}{210} times \frac{2}{2}. Multiply \frac{x^{5}}{20} times \frac{21}{21}.

$$\frac{2\left(5x^{7}-7x^{6}\right)}{420}-\frac{21x^{5}}{420}+\frac{x^{4}}{12}+3x+2$$

Since \frac{2\left(5x^{7}-7x^{6}\right)}{420} and \frac{21x^{5}}{420} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(5x^{7}-7x^{6}\right)-21x^{5}}{420}+\frac{x^{4}}{12}+3x+2$$

Do the multiplications in 2\left(5x^{7}-7x^{6}\right)-21x^{5}.

$$\frac{10x^{7}-14x^{6}-21x^{5}}{420}+\frac{x^{4}}{12}+3x+2$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 420 and 12 is 420. Multiply \frac{x^{4}}{12} times \frac{35}{35}.

$$\frac{10x^{7}-14x^{6}-21x^{5}}{420}+\frac{35x^{4}}{420}+3x+2$$

Since \frac{10x^{7}-14x^{6}-21x^{5}}{420} and \frac{35x^{4}}{420} have the same denominator, add them by adding their numerators.

$$\frac{10x^{7}-14x^{6}-21x^{5}+35x^{4}}{420}+3x+2$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3x+2 times \frac{420}{420}.

$$\frac{10x^{7}-14x^{6}-21x^{5}+35x^{4}}{420}+\frac{420\left(3x+2\right)}{420}$$

Since \frac{10x^{7}-14x^{6}-21x^{5}+35x^{4}}{420} and \frac{420\left(3x+2\right)}{420} have the same denominator, add them by adding their numerators.

$$\frac{10x^{7}-14x^{6}-21x^{5}+35x^{4}+420\left(3x+2\right)}{420}$$

Do the multiplications in 10x^{7}-14x^{6}-21x^{5}+35x^{4}+420\left(3x+2\right).

$$\frac{10x^{7}-14x^{6}-21x^{5}+35x^{4}+1260x+840}{420}$$

Factor

$$\frac{10x^{7}-14x^{6}-21x^{5}+35x^{4}+1260x+840}{420}$$

Solution Steps

Factor out \frac{1}{420}. Polynomial 10x^{7}-14x^{6}-21x^{5}+35x^{4}+1260x+840 is not factored since it does not have any rational roots.

$$\frac{10x^{7}-14x^{6}-21x^{5}+35x^{4}+1260x+840}{420}$$