tangent of (x^3+2x)^3,at x=4 — step-by-step solution

Find the tangent at

x = 4

Let's differentiate:

\frac{d}{d x} ({(x^{3} + 2 x)}^{3})

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

with f =

x^{3} + 2 x

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({(x^{3} + 2 x)}^{3})

3 \cdot \frac{d}{d x} (x^{3} + 2 x) \cdot {(x^{3} + 2 x)}^{2}

3 \cdot \frac{d}{d x} (x^{3} + 2 x) \cdot {(x^{3} + 2 x)}^{2}

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (x^{3} + 2 x)

\frac{d}{d x} (x^{3}) + \frac{d}{d x} (2 x)

3 \cdot (\frac{d}{d x} (x^{3}) + \frac{d}{d x} (2 x)) \cdot {(x^{3} + 2 x)}^{2}

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{3})

3 x^{2}

■A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (2 x)

2

3 \cdot (3 x^{2} + 2) \cdot {(x^{3} + 2 x)}^{2}

Compute f(

4

)

{(4^{3} + 2 \cdot 4)}^{3}

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

4^{3}

4 \cdot 4 \cdot 4

64

■Multiplications are done before additions and subtractions:

2 \cdot 4

8

{(64 + 8)}^{3}

ⓘ Parentheses are computed first:

64 + 8

72

72^{3}

ⓘ An integer power is a repeated product:

72^{3}

72 \cdot 72 \cdot 72

373,248

373,248

The point of tangency is (

4

373,248

)

Compute f'(

4

)

3 \cdot (3 \cdot 4^{2} + 2) \cdot {(4^{3} + 2 \cdot 4)}^{2}

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

4^{2}

4 \cdot 4

16

4^{3}

4 \cdot 4 \cdot 4

64

■Multiplications are done before additions and subtractions:

2 \cdot 4

8

3 \cdot (3 \cdot 16 + 2) \cdot {(64 + 8)}^{2}

ⓘ Multiplications are done before additions and subtractions:

3 \cdot 16

3 \cdot (10 + 6)

3 \cdot 10 + 3 \cdot 6

30 + 18

48

■Parentheses are computed first:

64 + 8

72

3 \cdot (48 + 2) \cdot 72^{2}

ⓘ Parentheses are computed first:

48 + 2

50

■Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

72^{2}

72 \cdot 72

5184

3 \cdot 50 \cdot 5184

ⅰ

3 \cdot 50

150

150 \cdot 5184

ⅰ

150 \cdot 5184

(100 + 50) \cdot 5184

100 \cdot 5184 + 50 \cdot 5184

518,400 + 259,200

777,600

777,600

The slope is f'(

4

) =

777,600

The tangent passes through this point with this slope:

y = 777,600 \cdot (x - 4) + 373,248

Compute the y-intercept:

b = 373,248 - 777,600 \cdot 4

373,248 - 777,600 \cdot 4

ⓘ Multiplications are done before additions and subtractions:

777,600 \cdot 4

3,110,400

373,248 - 3,110,400

ⓘ Start from

373,248

and move back

3,110,400

on the number line, going below zero:

373,248 - 3,110,400

-2,737,152

-2,737,152

▶Tangent line:

y = 777,600 x - 2,737,152