Calculus Examples

$y = x^{2} \sin^{4} (x) + x \cos^{- 3} (x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{2} \sin^{4} (x) + x \cos^{- 3} (x))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $x^{2} \sin^{4} (x) + x \cos^{- 3} (x)$ with respect to $x$ is $\frac{d}{d x} [x^{2} \sin^{4} (x)] + \frac{d}{d x} [x \cos^{- 3} (x)]$ .

$\frac{d}{d x} [x^{2} \sin^{4} (x)] + \frac{d}{d x} [x \cos^{- 3} (x)]$

Step 3.2

Evaluate $\frac{d}{d x} [x^{2} \sin^{4} (x)]$ .

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Step 3.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = \sin^{4} (x)$ .

$x^{2} \frac{d}{d x} [\sin^{4} (x)] + \sin^{4} (x) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x \cos^{- 3} (x)]$

Step 3.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \sin (x)$ .

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Step 3.2.2.1

To apply the Chain Rule, set $u_{1}$ as $\sin (x)$ .

$x^{2} (\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d x} [\sin (x)]) + \sin^{4} (x) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x \cos^{- 3} (x)]$

Step 3.2.2.2

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

$x^{2} (4 {u_{1}}^{3} \frac{d}{d x} [\sin (x)]) + \sin^{4} (x) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x \cos^{- 3} (x)]$

Step 3.2.2.3

Replace all occurrences of $u_{1}$ with $\sin (x)$ .

$x^{2} (4 \sin^{3} (x) \frac{d}{d x} [\sin (x)]) + \sin^{4} (x) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x \cos^{- 3} (x)]$

Step 3.2.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$x^{2} (4 \sin^{3} (x) \cos (x)) + \sin^{4} (x) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x \cos^{- 3} (x)]$

Step 3.2.4

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

$x^{2} (4 \sin^{3} (x) \cos (x)) + \sin^{4} (x) (2 x) + \frac{d}{d x} [x \cos^{- 3} (x)]$

Step 3.3

Evaluate $\frac{d}{d x} [x \cos^{- 3} (x)]$ .

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Step 3.3.1

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

$x^{2} (4 \sin^{3} (x) \cos (x)) + \sin^{4} (x) (2 x) + x \frac{d}{d x} [\cos^{- 3} (x)] + \cos^{- 3} (x) \frac{d}{d x} [x]$

Step 3.3.2

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 $g (x) = \cos (x)$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u_{2}$ as $\cos (x)$ .

$x^{2} (4 \sin^{3} (x) \cos (x)) + \sin^{4} (x) (2 x) + x (\frac{d}{d u_{2}} [{u_{2}}^{- 3}] \frac{d}{d x} [\cos (x)]) + \cos^{- 3} (x) \frac{d}{d x} [x]$

Step 3.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = - 3$ .

$x^{2} (4 \sin^{3} (x) \cos (x)) + \sin^{4} (x) (2 x) + x (- 3 {u_{2}}^{- 4} \frac{d}{d x} [\cos (x)]) + \cos^{- 3} (x) \frac{d}{d x} [x]$

Step 3.3.2.3

Replace all occurrences of $u_{2}$ with $\cos (x)$ .

$x^{2} (4 \sin^{3} (x) \cos (x)) + \sin^{4} (x) (2 x) + x (- 3 \cos^{- 4} (x) \frac{d}{d x} [\cos (x)]) + \cos^{- 3} (x) \frac{d}{d x} [x]$

Step 3.3.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$x^{2} (4 \sin^{3} (x) \cos (x)) + \sin^{4} (x) (2 x) + x (- 3 \cos^{- 4} (x) (- \sin (x))) + \cos^{- 3} (x) \frac{d}{d x} [x]$

Step 3.3.4

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

$x^{2} (4 \sin^{3} (x) \cos (x)) + \sin^{4} (x) (2 x) + x (- 3 \cos^{- 4} (x) (- \sin (x))) + \cos^{- 3} (x) \cdot 1$

Step 3.3.5

Multiply $- 1$ by $- 3$ .

$x^{2} (4 \sin^{3} (x) \cos (x)) + \sin^{4} (x) (2 x) + x (3 \cos^{- 4} (x) \sin (x)) + \cos^{- 3} (x) \cdot 1$

Step 3.3.6

Multiply $\cos^{- 3} (x)$ by $1$ .

$x^{2} (4 \sin^{3} (x) \cos (x)) + \sin^{4} (x) (2 x) + x (3 \cos^{- 4} (x) \sin (x)) + \cos^{- 3} (x)$

Step 3.4

Simplify.

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Step 3.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x^{2} (4 \sin^{3} (x) \cos (x)) + \sin^{4} (x) (2 x) + x (3 \frac{1}{\cos^{4} (x)} \sin (x)) + \cos^{- 3} (x)$

Step 3.4.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x^{2} (4 \sin^{3} (x) \cos (x)) + \sin^{4} (x) (2 x) + x (3 \frac{1}{\cos^{4} (x)} \sin (x)) + \frac{1}{\cos^{3} (x)}$

Step 3.4.3

Combine terms.

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Step 3.4.3.1

Convert from $\frac{1}{\cos^{4} (x)}$ to $\sec^{4} (x)$ .

$x^{2} (4 \sin^{3} (x) \cos (x)) + \sin^{4} (x) (2 x) + x (3 \sec^{4} (x) \sin (x)) + \frac{1}{\cos^{3} (x)}$

Step 3.4.3.2

Convert from $\frac{1}{\cos^{3} (x)}$ to $\sec^{3} (x)$ .

$x^{2} (4 \sin^{3} (x) \cos (x)) + \sin^{4} (x) (2 x) + x (3 \sec^{4} (x) \sin (x)) + \sec^{3} (x)$

Step 3.4.4

Reorder terms.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + 3 x \sec^{4} (x) \sin (x) + \sec^{3} (x)$

Step 3.4.5

Simplify each term.

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Step 3.4.5.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + 3 x {(\frac{1}{\cos (x)})}^{4} \sin (x) + \sec^{3} (x)$

Step 3.4.5.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + 3 x \frac{1^{4}}{\cos^{4} (x)} \sin (x) + \sec^{3} (x)$

Step 3.4.5.3

One to any power is one.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + 3 x \frac{1}{\cos^{4} (x)} \sin (x) + \sec^{3} (x)$

Step 3.4.5.4

Multiply $3 x \frac{1}{\cos^{4} (x)}$ .

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Step 3.4.5.4.1

Combine $3$ and $\frac{1}{\cos^{4} (x)}$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + x \frac{3}{\cos^{4} (x)} \sin (x) + \sec^{3} (x)$

Step 3.4.5.4.2

Combine $x$ and $\frac{3}{\cos^{4} (x)}$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{x \cdot 3}{\cos^{4} (x)} \sin (x) + \sec^{3} (x)$

Step 3.4.5.5

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

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 \cdot x}{\cos^{4} (x)} \sin (x) + \sec^{3} (x)$

Step 3.4.5.6

Combine $\frac{3 x}{\cos^{4} (x)}$ and $\sin (x)$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x \sin (x)}{\cos^{4} (x)} + \sec^{3} (x)$

Step 3.4.5.7

Rewrite $\sec (x)$ in terms of sines and cosines.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x \sin (x)}{\cos^{4} (x)} + {(\frac{1}{\cos (x)})}^{3}$

Step 3.4.5.8

Apply the product rule to $\frac{1}{\cos (x)}$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x \sin (x)}{\cos^{4} (x)} + \frac{1^{3}}{\cos^{3} (x)}$

Step 3.4.5.9

One to any power is one.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x \sin (x)}{\cos^{4} (x)} + \frac{1}{\cos^{3} (x)}$

Step 3.4.6

Simplify each term.

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Step 3.4.6.1

Factor $\cos (x)$ out of $\cos^{4} (x)$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x \sin (x)}{\cos (x) \cos^{3} (x)} + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.2

Separate fractions.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{\cos^{3} (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{\cos^{3} (x)} \tan (x) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.4

Combine $\frac{3 x}{\cos^{3} (x)}$ and $\tan (x)$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x \tan (x)}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.5

Factor $\cos (x)$ out of $\cos^{3} (x)$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x \tan (x)}{\cos (x) \cos^{2} (x)} + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.6

Separate fractions.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{\cos^{2} (x)} \cdot \frac{\tan (x)}{\cos (x)} + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.7

Rewrite $\tan (x)$ in terms of sines and cosines.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{\cos^{2} (x)} \cdot \frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)} + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.8

Rewrite $\frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)}$ as a product.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{\cos^{2} (x)} (\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.9

Simplify.

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Step 3.4.6.9.1

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{\cos^{2} (x)} (\tan (x) \frac{1}{\cos (x)}) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.9.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{\cos^{2} (x)} (\tan (x) \sec (x)) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.10

Multiply $\frac{3 x}{\cos^{2} (x)} (\tan (x) \sec (x))$ .

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Step 3.4.6.10.1

Combine $\tan (x)$ and $\frac{3 x}{\cos^{2} (x)}$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{\tan (x) (3 x)}{\cos^{2} (x)} \sec (x) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.10.2

Combine $\frac{\tan (x) (3 x)}{\cos^{2} (x)}$ and $\sec (x)$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{\tan (x) (3 x) \sec (x)}{\cos^{2} (x)} + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.11

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{\tan (x) (3 x) \sec (x)}{\cos (x) \cos (x)} + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.12

Separate fractions.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{(3 x) \sec (x)}{\cos (x)} \cdot \frac{\tan (x)}{\cos (x)} + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.13

Rewrite $\tan (x)$ in terms of sines and cosines.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{(3 x) \sec (x)}{\cos (x)} \cdot \frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)} + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.14

Rewrite $\frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)}$ as a product.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{(3 x) \sec (x)}{\cos (x)} (\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.15

Simplify.

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Step 3.4.6.15.1

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{(3 x) \sec (x)}{\cos (x)} (\tan (x) \frac{1}{\cos (x)}) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.15.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{(3 x) \sec (x)}{\cos (x)} (\tan (x) \sec (x)) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.16

Separate fractions.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{1} \cdot \frac{\sec (x)}{\cos (x)} (\tan (x) \sec (x)) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.17

Rewrite $\sec (x)$ in terms of sines and cosines.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{1} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)} (\tan (x) \sec (x)) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.18

Rewrite $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ as a product.

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{1} (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)}) (\tan (x) \sec (x)) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.19

Simplify.

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Step 3.4.6.19.1

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{1} (\sec (x) \frac{1}{\cos (x)}) (\tan (x) \sec (x)) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.19.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{1} (\sec (x) \sec (x)) (\tan (x) \sec (x)) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.19.3

Raise $\sec (x)$ to the power of $1$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{1} (\sec^{1} (x) \sec (x)) (\tan (x) \sec (x)) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.19.4

Raise $\sec (x)$ to the power of $1$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{1} (\sec^{1} (x) \sec^{1} (x)) (\tan (x) \sec (x)) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.19.5

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

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{1} {\sec (x)}^{1 + 1} (\tan (x) \sec (x)) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.19.6

Add $1$ and $1$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{1} \sec^{2} (x) (\tan (x) \sec (x)) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.20

Multiply $\sec^{2} (x)$ by $\sec (x)$ by adding the exponents.

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Step 3.4.6.20.1

Move $\sec (x)$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{1} (\sec (x) \sec^{2} (x)) \tan (x) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.20.2

Multiply $\sec (x)$ by $\sec^{2} (x)$ .

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Step 3.4.6.20.2.1

Raise $\sec (x)$ to the power of $1$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{1} (\sec^{1} (x) \sec^{2} (x)) \tan (x) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.20.2.2

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

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{1} {\sec (x)}^{1 + 2} \tan (x) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.20.3

Add $1$ and $2$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + \frac{3 x}{1} \sec^{3} (x) \tan (x) + \frac{1}{\cos^{3} (x)}$

Step 3.4.6.21

Divide $3 x$ by $1$ .

$4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + 3 x \sec^{3} (x) \tan (x) + \frac{1}{\cos^{3} (x)}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + 3 x \sec^{3} (x) \tan (x) + \frac{1}{\cos^{3} (x)}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 4 x^{2} \sin^{3} (x) \cos (x) + 2 x \sin^{4} (x) + 3 x \sec^{3} (x) \tan (x) + \frac{1}{\cos^{3} (x)}$