Derive the following functions: 1) f(x)=x, 2) f(x)=x³, 3) f(x)=1/x, 4) f(x)=2x^(1/2), 5) f(x)=x², 6) f(x)=xⁿ…

Question

Derive the following functions: 1) f(x)=x, 2) f(x)=x³, 3) f(x)=1/x, 4) f(x)=2x^(1/2), 5) f(x)=x², 6) f(x)=xⁿ, 7) f(x)=x⁻², 8) f(x)=x¹⁵, 9) f(x)=x²⁴, 10) f(x)=x⁻¹¹, 11) f(x)=-ex⁻¹¹, 12) f(x)=4x²⁰, 13) f(x)=3/x⁷

ScanSolve AI Tutor · Accepted Answer

Here are the derivatives of the given functions:

1) $f(x) = x$

Rewrite $f(x)$ as $x^1$ .
Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .
$f'(x) = 1 \cdot x^{1-1} = 1 \cdot x^0 = 1$ $\boxed{f'(x) = 1}$
2) $f(x) = x^3$
Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .
$f'(x) = 3x^{3-1} = 3x^2$ $\boxed{f'(x) = 3x^2}$
3) $f(x) = \frac{1}{x}$
Rewrite $f(x)$ as $x^{-1}$ .
Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .
$f'(x) = -1 \cdot x^{-1-1} = -x^{-2}$
Rewrite with a positive exponent.
$f'(x) = -\frac{1}{x^2}$ $\boxed{f'(x) = -\frac{1}{x^2}}$
4) $f(x) = 2\sqrt{x}$
Rewrite $f(x)$ as $2x^{1/2}$ .
Apply the power rule $\frac{d}{dx}(ax^n) = anx^{n-1}$ .
$f'(x) = 2 \cdot \frac{1}{2} x^{\frac{1}{2}-1} = 1 \cdot x^{-\frac{1}{2}}$
Rewrite with a positive exponent and radical form.
$f'(x) = \frac{1}{\sqrt{x}}$ $\boxed{f'(x) = \frac{1}{\sqrt{x}}}$
5) $f(x) = x^2$
Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .
$f'(x) = 2x^{2-1} = 2x$ $\boxed{f'(x) = 2x}$
6) $f(x) = x^n$
Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .
$f'(x) = nx^{n-1}$ $\boxed{f'(x) = nx^{n-1}}$
7) $f(x) = x^{-2}$
Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .
$f'(x) = -2x^{-2-1} = -2x^{-3}$
Rewrite with a positive exponent.
$f'(x) = -\frac{2}{x^3}$ $\boxed{f'(x) = -\frac{2}{x^3}}$
8) $f(x) = x^{15}$
Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .
$f'(x) = 15x^{15-1} = 15x^{14}$ $\boxed{f'(x) = 15x^{14}}$
9) $f(x) = x^{24}$
Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .
$f'(x) = 24x^{24-1} = 24x^{23}$ $\boxed{f'(x) = 24x^{23}}$
10) $f(x) = x^{-11}$
Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .
$f'(x) = -11x^{-11-1} = -11x^{-12}$
Rewrite with a positive exponent.
$f'(x) = -\frac{11}{x^{12}}$ $\boxed{f'(x) = -\frac{11}{x^{12}}}$
11) $f(x) = -2x^{11}$
Apply the power rule $\frac{d}{dx}(ax^n) = anx^{n-1}$ .
$f'(x) = -2 \cdot 11x^{11-1} = -22x^{10}$ $\boxed{f'(x) = -22x^{10}}$
12) $f(x) = 4x^{20}$
Apply the power rule $\frac{d}{dx}(ax^n) = anx^{n-1}$ .
$f'(x) = 4 \cdot 20x^{20-1} = 80x^{19}$ $\boxed{f'(x) = 80x^{19}}$
13) $f(x) = \frac{3}{x^7}$
Rewrite $f(x)$ as $3x^{-7}$ .
Apply the power rule $\frac{d}{dx}(ax^n) = anx^{n-1}$ .
$f'(x) = 3 \cdot (-7)x^{-7-1} = -21x^{-8}$
Rewrite with a positive exponent.
$f'(x) = -\frac{21}{x^8}$ $\boxed{f'(x) = -\frac{21}{x^8}}$

Derive the following functions: 1) f(x)=x, 2) f(x)=x³, 3) f(x)=1/x, 4) f(x)=2x^(1/2), 5) f(x)=x², 6) f(x)=xⁿ, 7) f(x)=x⁻², 8) f(x)=x¹⁵, 9) f(x)=x²⁴, 10) f(x)=x⁻¹¹, 11) f(x)=-ex⁻¹¹, 12) f(x)=4x²⁰, 13) f(x)=3/x⁷

Step-by-step solution

Rewrite $f(x)$ as $x^1$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Rewrite $f(x)$ as $x^{-1}$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Rewrite with a positive exponent.

Rewrite $f(x)$ as $2x^{1/2}$ .

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

Rewrite with a positive exponent and radical form.

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Rewrite with a positive exponent.

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Rewrite with a positive exponent.

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

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

Rewrite $f(x)$ as $3x^{-7}$ .

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

Rewrite with a positive exponent.

Still stuck on this one?

More Mathematics Questions

Derive the following functions: 1) f(x)=x, 2) f(x)=x³, 3) f(x)=1/x, 4) f(x)=2x^(1/2), 5) f(x)=x², 6) f(x)=xⁿ, 7) f(x)=x⁻², 8) f(x)=x¹⁵, 9) f(x)=x²⁴, 10) f(x)=x⁻¹¹, 11) f(x)=-ex⁻¹¹, 12) f(x)=4x²⁰, 13) f(x)=3/x⁷

Step-by-step solution

Rewrite f(x)f(x)f(x) as x1x^1x1.

Apply the power rule ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}dxd​(xn)=nxn−1.

Apply the power rule ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}dxd​(xn)=nxn−1.

Rewrite f(x)f(x)f(x) as x−1x^{-1}x−1.

Apply the power rule ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}dxd​(xn)=nxn−1.

Rewrite with a positive exponent.

Rewrite f(x)f(x)f(x) as 2x1/22x^{1/2}2x1/2.

Apply the power rule ddx(axn)=anxn−1\frac{d}{dx}(ax^n) = anx^{n-1}dxd​(axn)=anxn−1.

Rewrite with a positive exponent and radical form.

Apply the power rule ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}dxd​(xn)=nxn−1.

Apply the power rule ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}dxd​(xn)=nxn−1.

Apply the power rule ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}dxd​(xn)=nxn−1.

Rewrite with a positive exponent.

Apply the power rule ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}dxd​(xn)=nxn−1.

Apply the power rule ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}dxd​(xn)=nxn−1.

Apply the power rule ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}dxd​(xn)=nxn−1.

Rewrite with a positive exponent.

Apply the power rule ddx(axn)=anxn−1\frac{d}{dx}(ax^n) = anx^{n-1}dxd​(axn)=anxn−1.

Apply the power rule ddx(axn)=anxn−1\frac{d}{dx}(ax^n) = anx^{n-1}dxd​(axn)=anxn−1.

Rewrite f(x)f(x)f(x) as 3x−73x^{-7}3x−7.

Apply the power rule ddx(axn)=anxn−1\frac{d}{dx}(ax^n) = anx^{n-1}dxd​(axn)=anxn−1.

Rewrite with a positive exponent.

Still stuck on this one?

More Mathematics Questions

Rewrite $f(x)$ as $x^1$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Rewrite $f(x)$ as $x^{-1}$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Rewrite $f(x)$ as $2x^{1/2}$ .

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

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .

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

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

Rewrite $f(x)$ as $3x^{-7}$ .

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