Find the derivative of the given function.f(x)=2x3−1+lnxf(x)=2x^3-1+\ln xf(x)=2x3−1+lnx

6 x 2 + 1 x 6x^2+\frac{1}{x} 6 x 2 + x 1

Find the derivative of the given function. f ( x ) = 2 x 3 − 1 + ln x f(x)=2x^3-1+\ln x f ( x ) = 2 x 3 − 1 + ln x

we're asked to find the derivative of the function f of x equals 2x cubed minus 1 plus the natural log of x. Now since here, I just have a bunch of functions being added and subtracted, I can take each of their individual derivatives and add and subtract them as I should here. So in order to find my derivative here, f prime of x, we're starting with that first term, 2x cubed. I can find my derivative here using the power rule, pulling that 3 out front, multiplying by it, decreasing that power by 1. This gives me 6x squared. Now the derivative of a constant is 0, so I don't have to worry about anything there. If you want to, you can go ahead and indicate here a minus 0 for the derivative of minus 1. Then plus the derivative of the natural log of x, we know that this is just one over x. So rewriting this, this is just 6x squared plus 1 over x for this final derivative. Let us know if you have questions, and let's keep practicing.

Find the derivative of the given function. f ( x ) = 2 x 3 − 1 + ln ⁡ x f(x)=2x^3-1+\ln x f ( x ) = 2 x 3 − 1 + ln x

6 x 2 + 1 x 6x^2+\frac{1}{x} 6 x 2 + x 1

3 x 2 − 1 x \frac{3x^2-1}{x} x 3 x 2 − 1

4. Derivatives of Exponential & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Business Calculus

4. Derivatives of Exponential & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

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Multiple Choice

Find the derivative of the given function.
$f(x)=2x^3-1+\ln x$

$6x$

$\frac{3x^2-1}{x}$

$6x^2+\frac{1}{x}$

$6x-1$

Verified step by step guidance

Step 1: Break down the function into its components. The given function is f(x) = 2x^3 - 1 + ln(x). Each term will be differentiated separately using the rules of differentiation.

Step 2: Differentiate the first term, 2x^3, using the power rule. The power rule states that d/dx[x^n] = n*x^(n-1). For 2x^3, the derivative is 3*2*x^(3-1) = 6x^2.

Step 3: Differentiate the second term, -1. The derivative of a constant is always 0, so this term contributes nothing to the derivative.

Step 4: Differentiate the third term, ln(x), using the rule d/dx[ln(x)] = 1/x. The derivative of ln(x) is therefore 1/x.

Step 5: Combine the results from Steps 2, 3, and 4. The derivative of f(x) is f'(x) = 6x^2 + 1/x.

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