Find the first and second derivatives of the function.f(x)=x4x+5f'(x)=f''(x)=

Step 1f(x)=x4x+5f′(x)=ddx(x4x+5)=(ddx(x))(4x+5)−(ddx(4x+5))x(4x+5)2=(4x+5)−4x(4x+5)2=5(4x+5)2∴f′(x)=5(4x+5)2Step 2f″(x)=ddx(f′(x))=ddx(5(4x+5)2)=5ddx((4x+5)−2)=5×[−2(4x+5)−3ddx(4x+5)]=5[−2(4x+5)3⋅4]=−40(4x+5)3∴f″(x)=−40(4x+5)3

Step by step solution to "Find the first and second derivatives of the..."

fanyattehedzg

Answered question

2021-12-09

Find the first and second derivatives of the function.

f (x) = \frac{x}{4 x + 5}

f'(x)=
f''(x)=

Answer & Explanation

recoronarrv

Beginner2021-12-10Added 20 answers

Step 1

f (x) = \frac{x}{4 x + 5}

f^{'} (x) = \frac{d}{d x} (\frac{x}{4 x + 5})

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

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

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

∴ f^{'} (x) = \frac{5}{{(4 x + 5)}^{2}}

Step 2

f^{″} (x) = \frac{d}{d x} (f^{'} (x)) = \frac{d}{d x} (\frac{5}{{(4 x + 5)}^{2}})

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

= 5 \times [- 2 {(4 x + 5)}^{- 3} \frac{d}{d x} (4 x + 5)]

= 5 [- \frac{2}{{(4 x + 5)}^{3}} \cdot 4]

= - \frac{40}{{(4 x + 5)}^{3}}

∴ f^{″} (x) = - \frac{40}{{(4 x + 5)}^{3}}

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Calculus

What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders?
How do I find the y-intercept of a parabola?
What are the vertices of $9 x^{2} + 16 y^{2} = 144$ ?
How to determine the rate of change of a function?
Why are the tangents for 90 and 270 degrees undefined?
How to find the center and radius of the circle $x^{2} + y^{2} - 6 x + 8 y = 0$ ?
What is multiplicative inverse of a number?
How to find the continuity of a function on a closed interval?
How do I find the tangent line of a function?
How to find vertical asymptotes using limits?
How to find the center and radius of the circle $x^{2} - 12 x + y^{2} + 4 y + 15 = 0$ ?
Let f be a function so that (below). Which must be true?
I. f is continuous at x=2
II. f is differentiable at x=2
III. The derivative of f is continuous at x=2
(A) I (B) II (C) I and II (D) I and III (E) II and III
How to find the center and radius of the circle given $x^{2} + y^{2} + 8 x - 6 y = 0$ ?
How to find the center and radius of the circle $x^{2} + y^{2} + 4 x - 8 y + 4 = 0$ ?
How to identify the center and radius of the circle ${(x + 3)}^{2} + {(y - 8)}^{2} = 16$ ?

Didn't find what you were looking for?