tangent of f(x)=x^4-5x^2+2 at x=2 — step-by-step solution

Find the tangent at

x = 2

Let's perform the differentiation:

\frac{d}{d x} (x^{4} - 5 x^{2} + 2)

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

\frac{d}{d x} (x^{4} - 5 x^{2} + 2)

\frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 5 x^{2}) + 0

\frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 5 x^{2}) + 0

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{4})

4 x^{3}

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

2 x

■Adding 0 changes nothing:

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

\frac{d}{d x} (- 5 x^{2})

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

\frac{d}{d x} (- 5 x^{2})

-5 \cdot \frac{d}{d x} (x^{2})

4 x^{3} - 5 \cdot 2 x

ⓘ Multiplications are done before additions and subtractions:

5 \cdot 2

10

4 x^{3} - 10 x

Compute f(

2

)

2^{4} - 5 \cdot 2^{2} + 2

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

2^{4}

2 \cdot 2 \cdot 2 \cdot 2

16

2^{2}

2 \cdot 2

4

16 - 5 \cdot 4 + 2

ⓘ Multiplications are done before additions and subtractions:

5 \cdot 4

20

16 - 20 + 2

ⓘ Start from

16

and move back

20

on the number line, going below zero:

16 - 20

-4

- 4 + 2

ⓘ Start from

-4

and move forward

2

on the number line, getting closer to zero:

- 4 + 2

-2

-2

The point of tangency is (

2

-2

)

Compute f'(

2

)

4 \cdot 2^{3} - 10 \cdot 2

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

2^{3}

2 \cdot 2 \cdot 2

8

■Multiplications are done before additions and subtractions:

10 \cdot 2

20

4 \cdot 8 - 20

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 8

32

32 - 20

ⓘ Start from

32

and move back

20

on the number line:

32 - 20

12

12

The slope is f'(

2

) =

12

The tangent passes through this point with this slope:

y = 12 \cdot (x - 2) - 2

Compute the y-intercept:

b = - 2 - 12 \cdot 2

- 2 - 12 \cdot 2

ⓘ Multiplications are done before additions and subtractions:

12 \cdot 2

(10 + 2) \cdot 2

10 \cdot 2 + 2 \cdot 2

20 + 4

24

- 2 - 24

ⓘ Start from

-2

and move back

24

more on the number line, away from zero:

- 2 - 24

-26

-26

▶Tangent line:

y = 12 x - 26