tangent of 1/2 x^4-8 — step-by-step solution

⚠ 1/2 x^4-8 is read as (1/2)*x^4-8: use parentheses around the denominator to include the following factor in it

1) No x-coordinate is specified: find the tangent at the general point

x = a

, for real

a

⇥1.1) Let's perform the differentiation:

\frac{d}{d x} (\frac{x^{4}}{2} - 8)

⇥

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

⇥

\frac{1}{2} \cdot 4 x^{3}

⇥

2 x^{3}

Compute f(

a

)

The point of tangency is (

a

\frac{a^{4}}{2} - 8

)

Compute f'(

a

)

The slope is f'(

a

) =

2 a^{3}

The tangent passes through this point with this slope:

y = 2 a^{3} \cdot (x - a) + (\frac{a^{4}}{2} - 8)

Compute the y-intercept:

b = (\frac{a^{4}}{2} - 8) - 2 a^{3} \cdot a

⇥

(\frac{a^{4}}{2} - 8) - 2 a^{3} \cdot a

⇥

\frac{a^{4}}{2} - 8 - 2 a^{4}

⇥

\frac{-3 a^{4}}{2} - 8

⇥

- \frac{3 a^{4}}{2} - 8

⇥

y = 2 a^{3} \cdot x + (- \frac{3 a^{4}}{2} - 8)

▶Tangent line:

y = 2 a^{3} x - \frac{3 a^{4}}{2} - 8