Apply the differentiation rule d/dx( x) = 1/x.

Question

ScanSolve AI Tutor · Accepted Answer

Here are the differentiations for each function:

21. Differentiate $y = \ln x$

Apply the differentiation rule $\frac{d}{dx}(\ln x) = \frac{1}{x}$ .
$\frac{dy}{dx} = \frac{1}{x}$
The derivative is $\boxed{\frac{1}{x}}$ .

22. Differentiate $y = 5 \ln x$
Apply the constant multiple rule $\frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x))$ .
$\frac{dy}{dx} = 5 \frac{d}{dx}(\ln x)$
Differentiate $\ln x$ .
$\frac{dy}{dx} = 5 \cdot \frac{1}{x} = \frac{5}{x}$
The derivative is $\boxed{\frac{5}{x}}$ .

23. Differentiate $y = \ln x + 7$
Apply the sum rule $\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))$ and the rule for a constant $\frac{d}{dx}(c) = 0$ .
$\frac{dy}{dx} = \frac{d}{dx}(\ln x) + \frac{d}{dx}(7)$
Differentiate each term.
$\frac{dy}{dx} = \frac{1}{x} + 0 = \frac{1}{x}$
The derivative is $\boxed{\frac{1}{x}}$ .

24. Differentiate $y = 3 \ln x - 4$
Apply the difference rule and constant rules.
$\frac{dy}{dx} = \frac{d}{dx}(3 \ln x) - \frac{d}{dx}(4)$
Differentiate each term.
$\frac{dy}{dx} = 3 \cdot \frac{1}{x} - 0 = \frac{3}{x}$
The derivative is $\boxed{\frac{3}{x}}$ .

25. Differentiate $y = 8 \ln x + 5$
Apply the sum rule and constant rules.
$\frac{dy}{dx} = \frac{d}{dx}(8 \ln x) + \frac{d}{dx}(5)$
Differentiate each term.
$\frac{dy}{dx} = 8 \cdot \frac{1}{x} + 0 = \frac{8}{x}$
The derivative is $\boxed{\frac{8}{x}}$ .

26. Differentiate $y = 6 \ln x - 9$
Apply the difference rule and constant rules.
$\frac{dy}{dx} = \frac{d}{dx}(6 \ln x) - \frac{d}{dx}(9)$
Differentiate each term.
$\frac{dy}{dx} = 6 \cdot \frac{1}{x} - 0 = \frac{6}{x}$
The derivative is $\boxed{\frac{6}{x}}$ .

27. Differentiate $y = 10 \ln x + 3$
Apply the sum rule and constant rules.
$\frac{dy}{dx} = \frac{d}{dx}(10 \ln x) + \frac{d}{dx}(3)$
Differentiate each term.
$\frac{dy}{dx} = 10 \cdot \frac{1}{x} + 0 = \frac{10}{x}$
The derivative is $\boxed{\frac{10}{x}}$ .

28. Differentiate $y = 4 \ln x + 2 \ln x$
Simplify the expression.
$y = (4+2) \ln x = 6 \ln x$
Differentiate the simplified expression.
$\frac{dy}{dx} = \frac{d}{dx}(6 \ln x) = 6 \frac{d}{dx}(\ln x)$
Differentiate $\ln x$ .
$\frac{dy}{dx} = 6 \cdot \frac{1}{x} = \frac{6}{x}$
The derivative is $\boxed{\frac{6}{x}}$ .

29. Differentiate $y = 9 \ln x - 3 \ln x$
Simplify the expression.
$y = (9-3) \ln x = 6 \ln x$
Differentiate the simplified expression.
$\frac{dy}{dx} = \frac{d}{dx}(6 \ln x) = 6 \frac{d}{dx}(\ln x)$
Differentiate $\ln x$ .
$\frac{dy}{dx} = 6 \cdot \frac{1}{x} = \frac{6}{x}$
The derivative is $\boxed{\frac{6}{x}}$ .

30. Differentiate $y = 7 \ln x + 5$
Apply the sum rule and constant rules.
$\frac{dy}{dx} = \frac{d}{dx}(7 \ln x) + \frac{d}{dx}(5)$
Differentiate each term.
$\frac{dy}{dx} = 7 \cdot \frac{1}{x} + 0 = \frac{7}{x}$
The derivative is $\boxed{\frac{7}{x}}$ .

Apply the differentiation rule (d)/(dx)( x) = (1)/(x).

Step-by-step solution

Apply the differentiation rule $\frac{d}{dx}(\ln x) = \frac{1}{x}$ .

Apply the constant multiple rule $\frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x))$ .

Differentiate $\ln x$ .

Apply the sum rule $\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))$ and the rule for a constant $\frac{d}{dx}(c) = 0$ .

Differentiate each term.

Apply the difference rule and constant rules.

Differentiate each term.

Apply the sum rule and constant rules.

Differentiate each term.

Apply the difference rule and constant rules.

Differentiate each term.

Apply the sum rule and constant rules.

Differentiate each term.

Simplify the expression.

Differentiate the simplified expression.

Differentiate $\ln x$ .

Simplify the expression.

Differentiate the simplified expression.

Differentiate $\ln x$ .

Apply the sum rule and constant rules.

Differentiate each term.

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More Mathematics Questions

Apply the differentiation rule (d)/(dx)( x) = (1)/(x).

Step-by-step solution

Apply the differentiation rule ddx(ln⁡x)=1x\frac{d}{dx}(\ln x) = \frac{1}{x}dxd​(lnx)=x1​.

Apply the constant multiple rule ddx(c⋅f(x))=c⋅ddx(f(x))\frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x))dxd​(c⋅f(x))=c⋅dxd​(f(x)).

Differentiate ln⁡x\ln xlnx.

Apply the sum rule ddx(f(x)+g(x))=ddx(f(x))+ddx(g(x))\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))dxd​(f(x)+g(x))=dxd​(f(x))+dxd​(g(x)) and the rule for a constant ddx(c)=0\frac{d}{dx}(c) = 0dxd​(c)=0.

Differentiate each term.

Apply the difference rule and constant rules.

Differentiate each term.

Apply the sum rule and constant rules.

Differentiate each term.

Apply the difference rule and constant rules.

Differentiate each term.

Apply the sum rule and constant rules.

Differentiate each term.

Simplify the expression.

Differentiate the simplified expression.

Differentiate ln⁡x\ln xlnx.

Simplify the expression.

Differentiate the simplified expression.

Differentiate ln⁡x\ln xlnx.

Apply the sum rule and constant rules.

Differentiate each term.

Still stuck on this one?

More Mathematics Questions

Apply the differentiation rule $\frac{d}{dx}(\ln x) = \frac{1}{x}$ .

Apply the constant multiple rule $\frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x))$ .

Differentiate $\ln x$ .

Apply the sum rule $\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))$ and the rule for a constant $\frac{d}{dx}(c) = 0$ .

Differentiate $\ln x$ .

Differentiate $\ln x$ .