tangent of y=4e^x+x,(0,4) — step-by-step solution

⚠Ⓘ 4e is read as the product 4*e: for a power, write 4^e

1) Find the tangent at

x = 0

⇥1.1) Let's differentiate:

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

⇥

\frac{d}{d x} (4 e^{x}) + 1

⇥ℹ

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

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

‖

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

e^{x}

⇥

4 e^{x} + 1

Compute f(

0

)

⇥

4 e^{0} + 0

⇥

4

The point of tangency is (

0

4

)

Compute f'(

0

)

⇥

4 e^{0} + 1

⇥

4 + 1

⇥

5

The slope is f'(

0

) =

5

The tangent passes through this point with this slope:

y = 5 x + 4

Compute the y-intercept:

b = 4 - 5 \cdot 0

⇥

4 - 5 \cdot 0

⇥

4 + 0

⇥

4

▶Tangent line:

y = 5 x + 4