Find the derivative of the given function.g(t)=2t+log5tg(t)=2t+\log_5tg(t)=2t+log5t

2 + 1 t ln ⁡ 5 2+\frac{1}{t\ln5} 2 + t l n 5 1

Find the derivative of the given function. g ( t ) = 2 t + log 5 t g(t)=2t+\log_5t g ( t ) = 2 t + lo g 5 t

Here, we wanna find the derivative of the function g of t is equal to 2 t plus log base 5 of t. Now since I have 2 functions being added together, I can take each of their separate derivatives and add them together using my sum rule. So to find my derivative here, g prime of t, the derivative of 2 t is just 2. So this is 2 plus the derivative of a log base 5 of t. Now for a general logarithmic function, so a ddx of log base b of x, we know that this derivative is going to be 1 over x times the natural log of b. So here, when faced with a log base 5 of t, this is then going to be 1 over t times the natural log of that base, which is 5. So this is my full derivative here, g prime of t. I'll see you in the next one.

Find the derivative of the given function. g ( t ) = 2 t + log ⁡ 5 ⁡ t g(t)=2t+\log_5⁡t g ( t ) = 2 t + lo g 5 ⁡ t

2 + 1 t ln ⁡ 5 2+\frac{1}{t\ln5} 2 + t l n 5 1

2 t ln ⁡ 5 \frac{2}{t\ln5} t l n 5 2

1 t ln ⁡ 5 \frac{1}{t\ln5} t l n 5 1

2 + 5 ln ⁡ t 2+5\ln t 2 + 5 ln t

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Calculus

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

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Multiple Choice

Find the derivative of the given function.
$g(t)=2t+\log_5t$

$\frac{2}{t\ln5}$

$\frac{1}{t\ln5}$

$2+5\ln t$

$2+\frac{1}{t\ln5}$

Verified step by step guidance

Identify the function g(t) = 2t + log_5(t). This function is composed of two parts: a linear term 2t and a logarithmic term log_5(t).

Recall the derivative rules: The derivative of a constant times a variable, such as 2t, is simply the constant. The derivative of log_a(x) is 1/(x ln(a)), where a is the base of the logarithm.

Apply the derivative rule to the linear term: The derivative of 2t with respect to t is 2.

Apply the derivative rule to the logarithmic term: The derivative of log_5(t) with respect to t is 1/(t ln(5)).

Combine the derivatives of the individual terms: The derivative of g(t) is the sum of the derivatives of its parts, which is 2 + 1/(t ln(5)).

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