Find the Derivative Using Quotient Rule - d/dm (2m^4+21m^3+35m^2-37m+45)/(2m+7)

$\frac{(2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45)}{(2 m + 7)}$

Differentiate using the Quotient Rule which states that $\frac{d}{d m} [\frac{f (m)}{g (m)}]$ is $\frac{g (m) \frac{d}{d m} [f (m)] - f (m) \frac{d}{d m} [g (m)]}{{g (m)}^{2}}$ where $f (m) = 2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45$ and $g (m) = 2 m + 7$ .

$\frac{(2 m + 7) \frac{d}{d m} [2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45] - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

By the Sum Rule, the derivative of $2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45$ with respect to $m$ is $\frac{d}{d m} [2 m^{4}] + \frac{d}{d m} [21 m^{3}] + \frac{d}{d m} [35 m^{2}] + \frac{d}{d m} [- 37 m] + \frac{d}{d m} [45]$ .

$\frac{(2 m + 7) (\frac{d}{d m} [2 m^{4}] + \frac{d}{d m} [21 m^{3}] + \frac{d}{d m} [35 m^{2}] + \frac{d}{d m} [- 37 m] + \frac{d}{d m} [45]) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

Evaluate $\frac{d}{d m} [2 m^{4}]$ .

Since $2$ is constant with respect to $m$ , the derivative of $2 m^{4}$ with respect to $m$ is $2 \frac{d}{d m} [m^{4}]$ .

$\frac{(2 m + 7) (2 \frac{d}{d m} [m^{4}] + \frac{d}{d m} [21 m^{3}] + \frac{d}{d m} [35 m^{2}] + \frac{d}{d m} [- 37 m] + \frac{d}{d m} [45]) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

Differentiate using the Power Rule which states that $\frac{d}{d m} [m^{n}]$ is $n m^{n - 1}$ where $n = 4$ .

$\frac{(2 m + 7) (2 (4 m^{3}) + \frac{d}{d m} [21 m^{3}] + \frac{d}{d m} [35 m^{2}] + \frac{d}{d m} [- 37 m] + \frac{d}{d m} [45]) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

Multiply $4$ by $2$ .

$\frac{(2 m + 7) (8 m^{3} + \frac{d}{d m} [21 m^{3}] + \frac{d}{d m} [35 m^{2}] + \frac{d}{d m} [- 37 m] + \frac{d}{d m} [45]) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

Evaluate $\frac{d}{d m} [21 m^{3}]$ .

Since $21$ is constant with respect to $m$ , the derivative of $21 m^{3}$ with respect to $m$ is $21 \frac{d}{d m} [m^{3}]$ .

$\frac{(2 m + 7) (8 m^{3} + 21 \frac{d}{d m} [m^{3}] + \frac{d}{d m} [35 m^{2}] + \frac{d}{d m} [- 37 m] + \frac{d}{d m} [45]) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

Differentiate using the Power Rule which states that $\frac{d}{d m} [m^{n}]$ is $n m^{n - 1}$ where $n = 3$ .

$\frac{(2 m + 7) (8 m^{3} + 21 (3 m^{2}) + \frac{d}{d m} [35 m^{2}] + \frac{d}{d m} [- 37 m] + \frac{d}{d m} [45]) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

Multiply $3$ by $21$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + \frac{d}{d m} [35 m^{2}] + \frac{d}{d m} [- 37 m] + \frac{d}{d m} [45]) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

Evaluate $\frac{d}{d m} [35 m^{2}]$ .

Since $35$ is constant with respect to $m$ , the derivative of $35 m^{2}$ with respect to $m$ is $35 \frac{d}{d m} [m^{2}]$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 35 \frac{d}{d m} [m^{2}] + \frac{d}{d m} [- 37 m] + \frac{d}{d m} [45]) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

Differentiate using the Power Rule which states that $\frac{d}{d m} [m^{n}]$ is $n m^{n - 1}$ where $n = 2$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 35 (2 m) + \frac{d}{d m} [- 37 m] + \frac{d}{d m} [45]) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

Multiply $2$ by $35$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m + \frac{d}{d m} [- 37 m] + \frac{d}{d m} [45]) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

Evaluate $\frac{d}{d m} [- 37 m]$ .

Since $- 37$ is constant with respect to $m$ , the derivative of $- 37 m$ with respect to $m$ is $- 37 \frac{d}{d m} [m]$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m - 37 \frac{d}{d m} [m] + \frac{d}{d m} [45]) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

Differentiate using the Power Rule which states that $\frac{d}{d m} [m^{n}]$ is $n m^{n - 1}$ where $n = 1$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m - 37 \cdot 1 + \frac{d}{d m} [45]) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

Multiply $- 37$ by $1$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m - 37 + \frac{d}{d m} [45]) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

Differentiate.

Since $45$ is constant with respect to $m$ , the derivative of $45$ with respect to $m$ is $0$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m - 37 + 0) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

Add $8 m^{3} + 63 m^{2} + 70 m - 37$ and $0$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m - 37) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \frac{d}{d m} [2 m + 7]}{{(2 m + 7)}^{2}}$

By the Sum Rule, the derivative of $2 m + 7$ with respect to $m$ is $\frac{d}{d m} [2 m] + \frac{d}{d m} [7]$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m - 37) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) (\frac{d}{d m} [2 m] + \frac{d}{d m} [7])}{{(2 m + 7)}^{2}}$

Evaluate $\frac{d}{d m} [2 m]$ .

Since $2$ is constant with respect to $m$ , the derivative of $2 m$ with respect to $m$ is $2 \frac{d}{d m} [m]$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m - 37) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) (2 \frac{d}{d m} [m] + \frac{d}{d m} [7])}{{(2 m + 7)}^{2}}$

Differentiate using the Power Rule which states that $\frac{d}{d m} [m^{n}]$ is $n m^{n - 1}$ where $n = 1$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m - 37) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) (2 \cdot 1 + \frac{d}{d m} [7])}{{(2 m + 7)}^{2}}$

Multiply $2$ by $1$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m - 37) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) (2 + \frac{d}{d m} [7])}{{(2 m + 7)}^{2}}$

Differentiate using the Constant Rule.

Since $7$ is constant with respect to $m$ , the derivative of $7$ with respect to $m$ is $0$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m - 37) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) (2 + 0)}{{(2 m + 7)}^{2}}$

Add $2$ and $0$ .

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m - 37) - (2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \cdot 2}{{(2 m + 7)}^{2}}$

Simplify.

Apply the distributive property.

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m - 37) + (- (2 m^{4}) - (21 m^{3}) - (35 m^{2}) - (- 37 m) - 1 \cdot 45) \cdot 2}{{(2 m + 7)}^{2}}$

Apply the distributive property.

$\frac{(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m - 37) - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Simplify the numerator.

Simplify each term.

Expand $(2 m + 7) (8 m^{3} + 63 m^{2} + 70 m - 37)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{2 m (8 m^{3}) + 2 m (63 m^{2}) + 2 m (70 m) + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Simplify each term.

Rewrite using the commutative property of multiplication.

$\frac{2 \cdot 8 m \cdot m^{3} + 2 m (63 m^{2}) + 2 m (70 m) + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $m$ by $m^{3}$ by adding the exponents.

Move $m^{3}$ .

$\frac{2 \cdot 8 (m^{3} m) + 2 m (63 m^{2}) + 2 m (70 m) + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $m^{3}$ by $m$ .

Raise $m$ to the power of $1$ .

$\frac{2 \cdot 8 (m^{3} m^{1}) + 2 m (63 m^{2}) + 2 m (70 m) + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \cdot 8 m^{3 + 1} + 2 m (63 m^{2}) + 2 m (70 m) + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Add $3$ and $1$ .

$\frac{2 \cdot 8 m^{4} + 2 m (63 m^{2}) + 2 m (70 m) + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $2$ by $8$ .

$\frac{16 m^{4} + 2 m (63 m^{2}) + 2 m (70 m) + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{16 m^{4} + 2 \cdot 63 m \cdot m^{2} + 2 m (70 m) + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $m$ by $m^{2}$ by adding the exponents.

Move $m^{2}$ .

$\frac{16 m^{4} + 2 \cdot 63 (m^{2} m) + 2 m (70 m) + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $m^{2}$ by $m$ .

Raise $m$ to the power of $1$ .

$\frac{16 m^{4} + 2 \cdot 63 (m^{2} m^{1}) + 2 m (70 m) + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{16 m^{4} + 2 \cdot 63 m^{2 + 1} + 2 m (70 m) + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Add $2$ and $1$ .

$\frac{16 m^{4} + 2 \cdot 63 m^{3} + 2 m (70 m) + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $2$ by $63$ .

$\frac{16 m^{4} + 126 m^{3} + 2 m (70 m) + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{16 m^{4} + 126 m^{3} + 2 \cdot 70 m \cdot m + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $m$ by $m$ by adding the exponents.

Move $m$ .

$\frac{16 m^{4} + 126 m^{3} + 2 \cdot 70 (m \cdot m) + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $m$ by $m$ .

$\frac{16 m^{4} + 126 m^{3} + 2 \cdot 70 m^{2} + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $2$ by $70$ .

$\frac{16 m^{4} + 126 m^{3} + 140 m^{2} + 2 m \cdot - 37 + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $- 37$ by $2$ .

$\frac{16 m^{4} + 126 m^{3} + 140 m^{2} - 74 m + 7 (8 m^{3}) + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $8$ by $7$ .

$\frac{16 m^{4} + 126 m^{3} + 140 m^{2} - 74 m + 56 m^{3} + 7 (63 m^{2}) + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $63$ by $7$ .

$\frac{16 m^{4} + 126 m^{3} + 140 m^{2} - 74 m + 56 m^{3} + 441 m^{2} + 7 (70 m) + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $70$ by $7$ .

$\frac{16 m^{4} + 126 m^{3} + 140 m^{2} - 74 m + 56 m^{3} + 441 m^{2} + 490 m + 7 \cdot - 37 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $7$ by $- 37$ .

$\frac{16 m^{4} + 126 m^{3} + 140 m^{2} - 74 m + 56 m^{3} + 441 m^{2} + 490 m - 259 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Add $126 m^{3}$ and $56 m^{3}$ .

$\frac{16 m^{4} + 182 m^{3} + 140 m^{2} - 74 m + 441 m^{2} + 490 m - 259 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Add $140 m^{2}$ and $441 m^{2}$ .

$\frac{16 m^{4} + 182 m^{3} + 581 m^{2} - 74 m + 490 m - 259 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Add $- 74 m$ and $490 m$ .

$\frac{16 m^{4} + 182 m^{3} + 581 m^{2} + 416 m - 259 - (2 m^{4}) \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $2$ by $- 1$ .

$\frac{16 m^{4} + 182 m^{3} + 581 m^{2} + 416 m - 259 - 2 m^{4} \cdot 2 - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $2$ by $- 2$ .

$\frac{16 m^{4} + 182 m^{3} + 581 m^{2} + 416 m - 259 - 4 m^{4} - (21 m^{3}) \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $21$ by $- 1$ .

$\frac{16 m^{4} + 182 m^{3} + 581 m^{2} + 416 m - 259 - 4 m^{4} - 21 m^{3} \cdot 2 - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $2$ by $- 21$ .

$\frac{16 m^{4} + 182 m^{3} + 581 m^{2} + 416 m - 259 - 4 m^{4} - 42 m^{3} - (35 m^{2}) \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $35$ by $- 1$ .

$\frac{16 m^{4} + 182 m^{3} + 581 m^{2} + 416 m - 259 - 4 m^{4} - 42 m^{3} - 35 m^{2} \cdot 2 - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $2$ by $- 35$ .

$\frac{16 m^{4} + 182 m^{3} + 581 m^{2} + 416 m - 259 - 4 m^{4} - 42 m^{3} - 70 m^{2} - (- 37 m) \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $- 37$ by $- 1$ .

$\frac{16 m^{4} + 182 m^{3} + 581 m^{2} + 416 m - 259 - 4 m^{4} - 42 m^{3} - 70 m^{2} + 37 m \cdot 2 - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $2$ by $37$ .

$\frac{16 m^{4} + 182 m^{3} + 581 m^{2} + 416 m - 259 - 4 m^{4} - 42 m^{3} - 70 m^{2} + 74 m - 1 \cdot 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $- 1$ by $45$ .

$\frac{16 m^{4} + 182 m^{3} + 581 m^{2} + 416 m - 259 - 4 m^{4} - 42 m^{3} - 70 m^{2} + 74 m - 45 \cdot 2}{{(2 m + 7)}^{2}}$

Multiply $- 45$ by $2$ .

$\frac{16 m^{4} + 182 m^{3} + 581 m^{2} + 416 m - 259 - 4 m^{4} - 42 m^{3} - 70 m^{2} + 74 m - 90}{{(2 m + 7)}^{2}}$

Subtract $4 m^{4}$ from $16 m^{4}$ .

$\frac{12 m^{4} + 182 m^{3} + 581 m^{2} + 416 m - 259 - 42 m^{3} - 70 m^{2} + 74 m - 90}{{(2 m + 7)}^{2}}$

Subtract $42 m^{3}$ from $182 m^{3}$ .

$\frac{12 m^{4} + 140 m^{3} + 581 m^{2} + 416 m - 259 - 70 m^{2} + 74 m - 90}{{(2 m + 7)}^{2}}$

Subtract $70 m^{2}$ from $581 m^{2}$ .

$\frac{12 m^{4} + 140 m^{3} + 511 m^{2} + 416 m - 259 + 74 m - 90}{{(2 m + 7)}^{2}}$

Add $416 m$ and $74 m$ .

$\frac{12 m^{4} + 140 m^{3} + 511 m^{2} + 490 m - 259 - 90}{{(2 m + 7)}^{2}}$

Subtract $90$ from $- 259$ .

$\frac{12 m^{4} + 140 m^{3} + 511 m^{2} + 490 m - 349}{{(2 m + 7)}^{2}}$

Find the Derivative Using Quotient Rule - d/dm (2m^4+21m^3+35m^2-37m+45)/(2m+7)

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