tangent of y=x+sqrt(x),(49,56) — step-by-step solution

Find the tangent at

x = 49

Let's differentiate:

\frac{d}{d x} (x + \sqrt{x})

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

\frac{d}{d x} (x + \sqrt{x})

1 + \frac{d}{d x} (\sqrt{x})

1 + \frac{d}{d x} (\sqrt{x})

ⓘ The derivative of sqrt(x) is 1/(2*sqrt(x)):

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

\frac{1}{2 \sqrt{x}}

1 + \frac{1}{2 \sqrt{x}}

Compute f(

49

)

49 + \sqrt{49}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{49}

7

;

7^{2}

49

49 + 7

ⓘ Start from

49

and move forward

7

on the number line:

49 + 7

56

56

The point of tangency is (

49

56

)

Compute f'(

49

)

\frac{1}{2 \cdot \sqrt{49}} + 1

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{49}

7

;

7^{2}

49

\frac{1}{2 \cdot 7} + 1

ⅰ

2 \cdot 7

14

\frac{1}{14} + 1

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{1}{14} + 1

\frac{1 + 14}{14}

\frac{1 + 14}{14}

ⓘ The numerator and the denominator are computed before the division:

1 + 14

15

\frac{15}{14}

The slope is f'(

49

) =

\frac{15}{14}

The tangent passes through this point with this slope:

y = \frac{15}{14} \cdot (x - 49) + 56

Compute the y-intercept:

b = 56 - \frac{15}{14} \cdot 49

56 - \frac{15}{14} \cdot 49

ⓘ Multiplying a fraction by a number multiplies its numerator:

\frac{15}{14} \cdot 49

\frac{15 \cdot 49}{14}

\frac{735}{14}

56 - \frac{735}{14}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

7

, their greatest common divisor:

- \frac{735}{14}

- \frac{105 \cdot 7}{2 \cdot 7}

- \frac{105}{2}

56 - \frac{105}{2}

ⓘ To add fractions, put them over the same denominator, then add the numerators:

56 - \frac{105}{2}

\frac{2 \cdot 56 - 105}{2}

\frac{2 \cdot 56 - 105}{2}

ⓘ Multiplications are done before additions and subtractions:

2 \cdot 56

2 \cdot (50 + 6)

2 \cdot 50 + 2 \cdot 6

100 + 12

112

\frac{112 - 105}{2}

ⓘ The numerator and the denominator are computed before the division:

112 - 105

7

\frac{7}{2}

y = \frac{15}{14} \cdot x + \frac{7}{2}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{15}{14} \cdot x

\frac{15 x}{14}

■To add fractions, put them over the same denominator, then add the numerators:

\frac{15 x}{14} + \frac{7}{2}

\frac{15 x + 7 \cdot 7}{14}

y = \frac{15 x + 7 \cdot 7}{14}

ⓘ Multiplications are done before additions and subtractions:

7 \cdot 7

49

▶Tangent line:

y = \frac{15 x + 49}{14}