Find the Derivative Using Quotient Rule - d/dd (df(x))/(dx)=(d(5x^2-7)^3)/(dx)

$\frac{d f (x)}{d x} = \frac{d {(5 x^{2} - 7)}^{3}}{d x}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d \cdot d} [\frac{f (d)}{g (d)}]$ is $\frac{g (d) \frac{d}{d \cdot d} [f (d)] - f (d) \frac{d}{d \cdot d} [g (d)]}{{g (d)}^{2}}$ where $f (d) = d {(5 x^{2} - 7)}^{3}$ and $g (d) = d x$ .

$\frac{d x \frac{d}{d \cdot d} [d {(5 x^{2} - 7)}^{3}] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Step 2

Use the Binomial Theorem.

$\frac{d x \frac{d}{d \cdot d} [d ({(5 x^{2})}^{3} + 3 {(5 x^{2})}^{2} \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3})] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Step 3

Simplify each term.

Apply the product rule to $5 x^{2}$ .

$\frac{d x \frac{d}{d \cdot d} [d (5^{3} {(x^{2})}^{3} + 3 {(5 x^{2})}^{2} \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3})] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Raise $5$ to the power of $3$ .

$\frac{d x \frac{d}{d \cdot d} [d (125 {(x^{2})}^{3} + 3 {(5 x^{2})}^{2} \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3})] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Multiply the exponents in ${(x^{2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d x \frac{d}{d \cdot d} [d (125 x^{2 \cdot 3} + 3 {(5 x^{2})}^{2} \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3})] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Multiply $2$ by $3$ .

$\frac{d x \frac{d}{d \cdot d} [d (125 x^{6} + 3 {(5 x^{2})}^{2} \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3})] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Apply the product rule to $5 x^{2}$ .

$\frac{d x \frac{d}{d \cdot d} [d (125 x^{6} + 3 (5^{2} {(x^{2})}^{2}) \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3})] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Raise $5$ to the power of $2$ .

$\frac{d x \frac{d}{d \cdot d} [d (125 x^{6} + 3 (25 {(x^{2})}^{2}) \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3})] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d x \frac{d}{d \cdot d} [d (125 x^{6} + 3 (25 x^{2 \cdot 2}) \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3})] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Multiply $2$ by $2$ .

$\frac{d x \frac{d}{d \cdot d} [d (125 x^{6} + 3 (25 x^{4}) \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3})] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Multiply $25$ by $3$ .

$\frac{d x \frac{d}{d \cdot d} [d (125 x^{6} + 75 x^{4} \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3})] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Multiply $- 7$ by $75$ .

$\frac{d x \frac{d}{d \cdot d} [d (125 x^{6} - 525 x^{4} + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3})] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Multiply $5$ by $3$ .

$\frac{d x \frac{d}{d \cdot d} [d (125 x^{6} - 525 x^{4} + 15 x^{2} {(- 7)}^{2} + {(- 7)}^{3})] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Raise $- 7$ to the power of $2$ .

$\frac{d x \frac{d}{d \cdot d} [d (125 x^{6} - 525 x^{4} + 15 x^{2} \cdot 49 + {(- 7)}^{3})] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Multiply $49$ by $15$ .

$\frac{d x \frac{d}{d \cdot d} [d (125 x^{6} - 525 x^{4} + 735 x^{2} + {(- 7)}^{3})] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Raise $- 7$ to the power of $3$ .

$\frac{d x \frac{d}{d \cdot d} [d (125 x^{6} - 525 x^{4} + 735 x^{2} - 343)] - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Step 4

Since $125 x^{6} - 525 x^{4} + 735 x^{2} - 343$ is constant with respect to $d$ , the derivative of $d (125 x^{6} - 525 x^{4} + 735 x^{2} - 343)$ with respect to $d$ is $(125 x^{6} - 525 x^{4} + 735 x^{2} - 343) \frac{d}{d \cdot d} [d]$ .

$\frac{d x ((125 x^{6} - 525 x^{4} + 735 x^{2} - 343) \frac{d}{d \cdot d} [d]) - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Step 5

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

$\frac{d x ((125 x^{6} - 525 x^{4} + 735 x^{2} - 343) \cdot 1) - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Step 6

Multiply $125 x^{6} - 525 x^{4} + 735 x^{2} - 343$ by $1$ .

$\frac{d x (125 x^{6} - 525 x^{4} + 735 x^{2} - 343) - d {(5 x^{2} - 7)}^{3} \frac{d}{d \cdot d} [d x]}{{(d x)}^{2}}$

Step 7

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

$\frac{d x (125 x^{6} - 525 x^{4} + 735 x^{2} - 343) - d {(5 x^{2} - 7)}^{3} (x \frac{d}{d \cdot d} [d])}{{(d x)}^{2}}$

Step 8

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

$\frac{d x (125 x^{6} - 525 x^{4} + 735 x^{2} - 343) - d {(5 x^{2} - 7)}^{3} (x \cdot 1)}{{(d x)}^{2}}$

Step 9

Multiply $x$ by $1$ .

$\frac{d x (125 x^{6} - 525 x^{4} + 735 x^{2} - 343) - d {(5 x^{2} - 7)}^{3} x}{{(d x)}^{2}}$

Step 10

Simplify.

Apply the product rule to $d x$ .

$\frac{d x (125 x^{6} - 525 x^{4} + 735 x^{2} - 343) - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

Apply the distributive property.

$\frac{d x (125 x^{6}) + d x (- 525 x^{4}) + d x (735 x^{2}) + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

Simplify the numerator.

Simplify each term.

Rewrite using the commutative property of multiplication.

$\frac{125 d x \cdot x^{6} + d x (- 525 x^{4}) + d x (735 x^{2}) + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

Multiply $x$ by $x^{6}$ by adding the exponents.

Move $x^{6}$ .

$\frac{125 d (x^{6} x) + d x (- 525 x^{4}) + d x (735 x^{2}) + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

Multiply $x^{6}$ by $x$ .

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

$\frac{125 d (x^{6} x^{1}) + d x (- 525 x^{4}) + d x (735 x^{2}) + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

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

$\frac{125 d x^{6 + 1} + d x (- 525 x^{4}) + d x (735 x^{2}) + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

Add $6$ and $1$ .

$\frac{125 d x^{7} + d x (- 525 x^{4}) + d x (735 x^{2}) + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{125 d x^{7} - 525 d x \cdot x^{4} + d x (735 x^{2}) + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

Multiply $x$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$\frac{125 d x^{7} - 525 d (x^{4} x) + d x (735 x^{2}) + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

Multiply $x^{4}$ by $x$ .

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

$\frac{125 d x^{7} - 525 d (x^{4} x^{1}) + d x (735 x^{2}) + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

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

$\frac{125 d x^{7} - 525 d x^{4 + 1} + d x (735 x^{2}) + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

Add $4$ and $1$ .

$\frac{125 d x^{7} - 525 d x^{5} + d x (735 x^{2}) + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x \cdot x^{2} + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

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

Move $x^{2}$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d (x^{2} x) + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

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

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

$\frac{125 d x^{7} - 525 d x^{5} + 735 d (x^{2} x^{1}) + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

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

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{2 + 1} + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

Add $2$ and $1$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} + d x \cdot - 343 - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

Move $- 343$ to the left of $d x$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 \cdot (d x) - d {(5 x^{2} - 7)}^{3} x}{d^{2} x^{2}}$

Use the Binomial Theorem.

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d ({(5 x^{2})}^{3} + 3 {(5 x^{2})}^{2} \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3}) x}{d^{2} x^{2}}$

Simplify each term.

Apply the product rule to $5 x^{2}$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d (5^{3} {(x^{2})}^{3} + 3 {(5 x^{2})}^{2} \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3}) x}{d^{2} x^{2}}$

Raise $5$ to the power of $3$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d (125 {(x^{2})}^{3} + 3 {(5 x^{2})}^{2} \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3}) x}{d^{2} x^{2}}$

Multiply the exponents in ${(x^{2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d (125 x^{2 \cdot 3} + 3 {(5 x^{2})}^{2} \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3}) x}{d^{2} x^{2}}$

Multiply $2$ by $3$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d (125 x^{6} + 3 {(5 x^{2})}^{2} \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3}) x}{d^{2} x^{2}}$

Apply the product rule to $5 x^{2}$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d (125 x^{6} + 3 (5^{2} {(x^{2})}^{2}) \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3}) x}{d^{2} x^{2}}$

Raise $5$ to the power of $2$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d (125 x^{6} + 3 (25 {(x^{2})}^{2}) \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3}) x}{d^{2} x^{2}}$

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d (125 x^{6} + 3 (25 x^{2 \cdot 2}) \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3}) x}{d^{2} x^{2}}$

Multiply $2$ by $2$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d (125 x^{6} + 3 (25 x^{4}) \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3}) x}{d^{2} x^{2}}$

Multiply $25$ by $3$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d (125 x^{6} + 75 x^{4} \cdot - 7 + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3}) x}{d^{2} x^{2}}$

Multiply $- 7$ by $75$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d (125 x^{6} - 525 x^{4} + 3 (5 x^{2}) {(- 7)}^{2} + {(- 7)}^{3}) x}{d^{2} x^{2}}$

Multiply $5$ by $3$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d (125 x^{6} - 525 x^{4} + 15 x^{2} {(- 7)}^{2} + {(- 7)}^{3}) x}{d^{2} x^{2}}$

Raise $- 7$ to the power of $2$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d (125 x^{6} - 525 x^{4} + 15 x^{2} \cdot 49 + {(- 7)}^{3}) x}{d^{2} x^{2}}$

Multiply $49$ by $15$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d (125 x^{6} - 525 x^{4} + 735 x^{2} + {(- 7)}^{3}) x}{d^{2} x^{2}}$

Raise $- 7$ to the power of $3$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - d (125 x^{6} - 525 x^{4} + 735 x^{2} - 343) x}{d^{2} x^{2}}$

Apply the distributive property.

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x + (- d (125 x^{6}) - d (- 525 x^{4}) - d (735 x^{2}) - d \cdot - 343) x}{d^{2} x^{2}}$

Simplify.

Rewrite using the commutative property of multiplication.

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x + (- 1 \cdot 125 d x^{6} - d (- 525 x^{4}) - d (735 x^{2}) - d \cdot - 343) x}{d^{2} x^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x + (- 1 \cdot 125 d x^{6} - 1 \cdot - 525 d x^{4} - d (735 x^{2}) - d \cdot - 343) x}{d^{2} x^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x + (- 1 \cdot 125 d x^{6} - 1 \cdot - 525 d x^{4} - 1 \cdot 735 d x^{2} - d \cdot - 343) x}{d^{2} x^{2}}$

Multiply $- 343$ by $- 1$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x + (- 1 \cdot 125 d x^{6} - 1 \cdot - 525 d x^{4} - 1 \cdot 735 d x^{2} + 343 d) x}{d^{2} x^{2}}$

Simplify each term.

Multiply $- 1$ by $125$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x + (- 125 d x^{6} - 1 \cdot - 525 d x^{4} - 1 \cdot 735 d x^{2} + 343 d) x}{d^{2} x^{2}}$

Multiply $- 1$ by $- 525$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x + (- 125 d x^{6} + 525 d x^{4} - 1 \cdot 735 d x^{2} + 343 d) x}{d^{2} x^{2}}$

Multiply $- 1$ by $735$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x + (- 125 d x^{6} + 525 d x^{4} - 735 d x^{2} + 343 d) x}{d^{2} x^{2}}$

Apply the distributive property.

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - 125 d x^{6} x + 525 d x^{4} x - 735 d x^{2} x + 343 d x}{d^{2} x^{2}}$

Simplify.

Multiply $x^{6}$ by $x$ by adding the exponents.

Move $x$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - 125 d (x \cdot x^{6}) + 525 d x^{4} x - 735 d x^{2} x + 343 d x}{d^{2} x^{2}}$

Multiply $x$ by $x^{6}$ .

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

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - 125 d (x^{1} x^{6}) + 525 d x^{4} x - 735 d x^{2} x + 343 d x}{d^{2} x^{2}}$

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

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - 125 d x^{1 + 6} + 525 d x^{4} x - 735 d x^{2} x + 343 d x}{d^{2} x^{2}}$

Add $1$ and $6$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - 125 d x^{7} + 525 d x^{4} x - 735 d x^{2} x + 343 d x}{d^{2} x^{2}}$

Multiply $x^{4}$ by $x$ by adding the exponents.

Move $x$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - 125 d x^{7} + 525 d (x \cdot x^{4}) - 735 d x^{2} x + 343 d x}{d^{2} x^{2}}$

Multiply $x$ by $x^{4}$ .

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

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - 125 d x^{7} + 525 d (x^{1} x^{4}) - 735 d x^{2} x + 343 d x}{d^{2} x^{2}}$

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

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - 125 d x^{7} + 525 d x^{1 + 4} - 735 d x^{2} x + 343 d x}{d^{2} x^{2}}$

Add $1$ and $4$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - 125 d x^{7} + 525 d x^{5} - 735 d x^{2} x + 343 d x}{d^{2} x^{2}}$

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

Move $x$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - 125 d x^{7} + 525 d x^{5} - 735 d (x \cdot x^{2}) + 343 d x}{d^{2} x^{2}}$

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

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

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - 125 d x^{7} + 525 d x^{5} - 735 d (x^{1} x^{2}) + 343 d x}{d^{2} x^{2}}$

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

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - 125 d x^{7} + 525 d x^{5} - 735 d x^{1 + 2} + 343 d x}{d^{2} x^{2}}$

Add $1$ and $2$ .

$\frac{125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - 125 d x^{7} + 525 d x^{5} - 735 d x^{3} + 343 d x}{d^{2} x^{2}}$

Combine the opposite terms in $125 d x^{7} - 525 d x^{5} + 735 d x^{3} - 343 d x - 125 d x^{7} + 525 d x^{5} - 735 d x^{3} + 343 d x$ .

Subtract $125 d x^{7}$ from $125 d x^{7}$ .

$\frac{- 525 d x^{5} + 735 d x^{3} - 343 d x + 0 + 525 d x^{5} - 735 d x^{3} + 343 d x}{d^{2} x^{2}}$

Add $- 525 d x^{5} + 735 d x^{3} - 343 d x$ and $0$ .

$\frac{- 525 d x^{5} + 735 d x^{3} - 343 d x + 525 d x^{5} - 735 d x^{3} + 343 d x}{d^{2} x^{2}}$

Add $- 525 d x^{5}$ and $525 d x^{5}$ .

$\frac{735 d x^{3} - 343 d x + 0 - 735 d x^{3} + 343 d x}{d^{2} x^{2}}$

Add $735 d x^{3} - 343 d x$ and $0$ .

$\frac{735 d x^{3} - 343 d x - 735 d x^{3} + 343 d x}{d^{2} x^{2}}$

Subtract $735 d x^{3}$ from $735 d x^{3}$ .

$\frac{- 343 d x + 0 + 343 d x}{d^{2} x^{2}}$

Add $- 343 d x$ and $0$ .

$\frac{- 343 d x + 343 d x}{d^{2} x^{2}}$

Add $- 343 d x$ and $343 d x$ .

$\frac{0}{d^{2} x^{2}}$

Divide $0$ by $d^{2} x^{2}$ .

$0$

Find the Derivative Using Quotient Rule - d/dd (df(x))/(dx)=(d(5x^2-7)^3)/(dx)

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