Find the derivative of the given function.f(x)=8log2xf(x)=8\log_2xf(x)=8log2x

8 x ln ⁡ 2 \frac{8}{x\ln2} x l n 2 8

Find the derivative of the given function. f ( x ) = 8 log 2 x f(x)=8\log_2x f ( x ) = 8 lo g 2 x

this problem, we wanna find the derivative of the function f of x equals 8 times log base 2 of x. Now for any general logarithmic function with some base b, we know that its derivative, so ddx of a log base b of x, is equal to 1 over x times the natural log of that base b. Now since here we just have a constant multiplying a general logarithmic function, that means that when finding our derivative here, f prime of x, we're gonna take that constant and just multiply it by this derivative. Now based on this rule, this derivative is then going to be 1 over x times the natural log of 2. Now we can rewrite this a little bit more compactly as 8 over x times the natural log of 2, and this is our final derivative here, f prime of x. Let us know if you have questions, and I'll see you in the next one.

Find the derivative of the given function. f ( x ) = 8 log ⁡ 2 ⁡ x f(x)=8\log_2⁡x f ( x ) = 8 lo g 2 ⁡ x

8 x ln ⁡ 2 \frac{8}{x\ln2} x l n 2 8

8 2 ln ⁡ x \frac{8}{2\ln x} 2 l n x 8

1 x ln ⁡ 8 \frac{1}{x\ln8} x l n 8 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.
$f(x)=8\log_2x$

$\frac{8}{2\ln x}$

$\frac{8}{x\ln2}$

$\frac{8}{x}$

$\frac{1}{x\ln8}$

Verified step by step guidance

Step 1: Recognize that the function is f(x) = 8 * log_2(x). The derivative of a logarithmic function with base 'a' is given by d/dx[log_a(x)] = 1 / (x * ln(a)), where ln(a) is the natural logarithm of the base.

Step 2: Apply the constant multiple rule of differentiation. Since the function is multiplied by 8, the derivative becomes f'(x) = 8 * d/dx[log_2(x)].

Step 3: Substitute the derivative formula for log_2(x). Using d/dx[log_2(x)] = 1 / (x * ln(2)), the expression becomes f'(x) = 8 * (1 / (x * ln(2))).

Step 4: Simplify the expression. Combine the constants and terms to get f'(x) = 8 / (x * ln(2)).

Step 5: The final simplified derivative is f'(x) = 8 / (x * ln(2)). This is the derivative of the given function.

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