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

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

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

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

Since $d$ is constant with respect to $x$ , the derivative of $d {(5 x^{2} - 7)}^{3}$ with respect to $x$ is $d \frac{d}{d x} [{(5 x^{2} - 7)}^{3}]$ .

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

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

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To apply the Chain Rule, set $u$ as $5 x^{2} - 7$ .

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

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

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

Replace all occurrences of $u$ with $5 x^{2} - 7$ .

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

Differentiate.

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Move $3$ to the left of $d$ .

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

By the Sum Rule, the derivative of $5 x^{2} - 7$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 7]$ .

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

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

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

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

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

Multiply $2$ by $5$ .

$\frac{d x (3 d {(5 x^{2} - 7)}^{2} (10 x + \frac{d}{d x} [- 7])) - d {(5 x^{2} - 7)}^{3} \frac{d}{d x} [d x]}{{(d x)}^{2}}$

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

$\frac{d x (3 d {(5 x^{2} - 7)}^{2} (10 x + 0)) - d {(5 x^{2} - 7)}^{3} \frac{d}{d x} [d x]}{{(d x)}^{2}}$

Simplify the expression.

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Add $10 x$ and $0$ .

$\frac{d x (3 d {(5 x^{2} - 7)}^{2} (10 x)) - d {(5 x^{2} - 7)}^{3} \frac{d}{d x} [d x]}{{(d x)}^{2}}$

Multiply $10$ by $3$ .

$\frac{d x (30 d {(5 x^{2} - 7)}^{2} x) - d {(5 x^{2} - 7)}^{3} \frac{d}{d x} [d x]}{{(d x)}^{2}}$

Reorder the factors of $30 d {(5 x^{2} - 7)}^{2} x$ .

$\frac{d x (30 d x {(5 x^{2} - 7)}^{2}) - d {(5 x^{2} - 7)}^{3} \frac{d}{d x} [d x]}{{(d x)}^{2}}$

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

$\frac{d x (30 d x {(5 x^{2} - 7)}^{2}) - d {(5 x^{2} - 7)}^{3} (d \frac{d}{d x} [x])}{{(d x)}^{2}}$

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

$\frac{d x (30 d x {(5 x^{2} - 7)}^{2}) - d {(5 x^{2} - 7)}^{3} (d \cdot 1)}{{(d x)}^{2}}$

Multiply $d$ by $1$ .

$\frac{d x (30 d x {(5 x^{2} - 7)}^{2}) - d {(5 x^{2} - 7)}^{3} d}{{(d x)}^{2}}$

Simplify.

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Apply the product rule to $d x$ .

$\frac{d x (30 d x {(5 x^{2} - 7)}^{2}) - d {(5 x^{2} - 7)}^{3} d}{d^{2} x^{2}}$

Simplify the numerator.

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Factor $d^{2} {(5 x^{2} - 7)}^{2}$ out of $d x (30 d x {(5 x^{2} - 7)}^{2}) - d {(5 x^{2} - 7)}^{3} d$ .

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Factor $d^{2} {(5 x^{2} - 7)}^{2}$ out of $d x (30 d x {(5 x^{2} - 7)}^{2})$ .

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{- 1} x (30 d x)) - d {(5 x^{2} - 7)}^{3} d}{d^{2} x^{2}}$

Factor $d^{2} {(5 x^{2} - 7)}^{2}$ out of $- d {(5 x^{2} - 7)}^{3} d$ .

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{- 1} x (30 d x)) + d^{2} {(5 x^{2} - 7)}^{2} (- d^{- 1} (5 x^{2} - 7) d)}{d^{2} x^{2}}$

Factor $d^{2} {(5 x^{2} - 7)}^{2}$ out of $d^{2} {(5 x^{2} - 7)}^{2} (d^{- 1} x (30 d x)) + d^{2} {(5 x^{2} - 7)}^{2} (- d^{- 1} (5 x^{2} - 7) d)$ .

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{- 1} x (30 d x) - d^{- 1} (5 x^{2} - 7) d)}{d^{2} x^{2}}$

Combine exponents.

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

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{1} d^{- 1} x (30 x) - d^{- 1} (5 x^{2} - 7) d)}{d^{2} x^{2}}$

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

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{1 - 1} x (30 x) - d^{- 1} (5 x^{2} - 7) d)}{d^{2} x^{2}}$

Subtract $1$ from $1$ .

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{0} x (30 x) - d^{- 1} (5 x^{2} - 7) d)}{d^{2} x^{2}}$

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

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{0} (30 (x^{1} x)) - d^{- 1} (5 x^{2} - 7) d)}{d^{2} x^{2}}$

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

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{0} (30 (x^{1} x^{1})) - d^{- 1} (5 x^{2} - 7) d)}{d^{2} x^{2}}$

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

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{0} (30 x^{1 + 1}) - d^{- 1} (5 x^{2} - 7) d)}{d^{2} x^{2}}$

Add $1$ and $1$ .

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{0} (30 x^{2}) - d^{- 1} (5 x^{2} - 7) d)}{d^{2} x^{2}}$

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

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{0} (30 x^{2}) - (d^{1} d^{- 1}) (5 x^{2} - 7))}{d^{2} x^{2}}$

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

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{0} (30 x^{2}) - d^{1 - 1} (5 x^{2} - 7))}{d^{2} x^{2}}$

Subtract $1$ from $1$ .

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{0} (30 x^{2}) - d^{0} (5 x^{2} - 7))}{d^{2} x^{2}}$

Factor $d^{0}$ out of $d^{0} (30 x^{2}) - d^{0} (5 x^{2} - 7)$ .

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Factor $d^{0}$ out of $- d^{0} (5 x^{2} - 7)$ .

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{0} (30 x^{2}) + d^{0} (- 1 (5 x^{2} - 7)))}{d^{2} x^{2}}$

Factor $d^{0}$ out of $d^{0} (30 x^{2}) + d^{0} (- 1 (5 x^{2} - 7))$ .

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{0} (30 x^{2} - 1 (5 x^{2} - 7)))}{d^{2} x^{2}}$

Apply the distributive property.

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{0} (30 x^{2} - 1 (5 x^{2}) - 1 \cdot - 7))}{d^{2} x^{2}}$

Multiply $5$ by $- 1$ .

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{0} (30 x^{2} - 5 x^{2} - 1 \cdot - 7))}{d^{2} x^{2}}$

Multiply $- 1$ by $- 7$ .

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{0} (30 x^{2} - 5 x^{2} + 7))}{d^{2} x^{2}}$

Subtract $5 x^{2}$ from $30 x^{2}$ .

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (d^{0} (25 x^{2} + 7))}{d^{2} x^{2}}$

Anything raised to $0$ is $1$ .

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (1 (25 x^{2} + 7))}{d^{2} x^{2}}$

Combine terms.

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Multiply $25 x^{2} + 7$ by $1$ .

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (25 x^{2} + 7)}{d^{2} x^{2}}$

Cancel the common factor of $d^{2}$ .

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Cancel the common factor.

$\frac{d^{2} {(5 x^{2} - 7)}^{2} (25 x^{2} + 7)}{d^{2} x^{2}}$

Rewrite the expression.

$\frac{{(5 x^{2} - 7)}^{2} (25 x^{2} + 7)}{x^{2}}$

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

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