What are the derivatives of the following functions: 2) y = a + e + pi + cos 14^circ + 2x 4) y = 2x(4x2…

Question

What are the derivatives of the following functions: 2) y = a + e + pi + cos 14^circ + 2x 4) y = 2x(4x2 + 2x − 6 + 2/x − 4/x2) 6) y = sin x + 2 cos x − 5 tan x?

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here are the derivatives for the ticked problems:

2) Given $y = a + e + \pi + \cos 14^\circ + 2x$

Identify constants.
The terms $a$ , $e$ , $\pi$ , and $\cos 14^\circ$ are all constants.
The derivative of any constant is 0.
Differentiate each term.
$\frac{dy}{dx} = \frac{d}{dx}(a) + \frac{d}{dx}(e) + \frac{d}{dx}(\pi) + \frac{d}{dx}(\cos 14^\circ) + \frac{d}{dx}(2x)$ $\frac{dy}{dx} = 0 + 0 + 0 + 0 + 2(1)x^{1-1}$ $\frac{dy}{dx} = 2x^0$ $\frac{dy}{dx} = 2$
The final answer is $\boxed{2}$ .

4) Given $y = 2x\left(4x^2 + 2x - 6 + \frac{2}{x} - \frac{4}{x^2}\right)$
Expand the expression by multiplying $2x$ by each term inside the parenthesis.
$y = 2x(4x^2) + 2x(2x) - 2x(6) + 2x\left(\frac{2}{x}\right) - 2x\left(\frac{4}{x^2}\right)$ $y = 8x^3 + 4x^2 - 12x + 4 - \frac{8}{x}$
Rewrite the term with a negative exponent.
$y = 8x^3 + 4x^2 - 12x + 4 - 8x^{-1}$
Differentiate each term using the power rule $\frac{d}{dx}(ax^n) = anx^{n-1}$ .
$\frac{dy}{dx} = 8(3)x^{3-1} + 4(2)x^{2-1} - 12(1)x^{1-1} + 0 - 8(-1)x^{-1-1}$ $\frac{dy}{dx} = 24x^2 + 8x - 12x^0 + 8x^{-2}$ $\frac{dy}{dx} = 24x^2 + 8x - 12 + \frac{8}{x^2}$
The final answer is $\boxed{24x^2 + 8x - 12 + \frac{8}{x^2}}$ .

6) Given $y = \sin x + 2 \cos x - 5 \tan x$
Differentiate each trigonometric term using standard derivative rules.
Recall:
$\frac{d}{dx}(\sin x) = \cos x$
$\frac{d}{dx}(\cos x) = -\sin x$
$\frac{d}{dx}(\tan x) = \sec^2 x$
Apply the derivative rules to each term.
$\frac{dy}{dx} = \frac{d}{dx}(\sin x) + 2\frac{d}{dx}(\cos x) - 5\frac{d}{dx}(\tan x)$ $\frac{dy}{dx} = \cos x + 2(-\sin x) - 5(\sec^2 x)$ $\frac{dy}{dx} = \cos x - 2\sin x - 5\sec^2 x$
The final answer is $\boxed{\cos x - 2\sin x - 5\sec^2 x}$ .

What are the derivatives of the following functions: 2) y = a + e + pi + cos 14^circ + 2x 4) y = 2x(4x2 + 2x − 6 + 2/x - 4/x2) 6) y = sin x + 2 cos x - 5 tan x?

Step-by-step solution

Identify constants.

Differentiate each term.

Expand the expression by multiplying $2x$ by each term inside the parenthesis.

Rewrite the term with a negative exponent.

Differentiate each term using the power rule $\frac{d}{dx}(ax^n) = anx^{n-1}$ .

Differentiate each trigonometric term using standard derivative rules.

Apply the derivative rules to each term.

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More Mathematics Questions

What are the derivatives of the following functions: 2) y = a + e + pi + cos 14^circ + 2x 4) y = 2x(4x2 + 2x − 6 + 2/x - 4/x2) 6) y = sin x + 2 cos x - 5 tan x?