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

4. Derivatives of Exponential & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Business Calculus

4. Derivatives of Exponential & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Struggling with Business Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

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

Step 1: Identify the function to differentiate. The given function is g(t) = 2t + log_5(t). This function consists of two terms: 2t and log_5(t).

Step 2: Differentiate the first term, 2t. The derivative of 2t with respect to t is simply 2, as the derivative of a linear term at is its coefficient.

Step 3: Differentiate the second term, log_5(t). Recall that the derivative of log_a(x) with respect to x is 1 / (x ln(a)), where a is the base of the logarithm. Here, the base is 5, so the derivative of log_5(t) is 1 / (t ln(5)).

Step 4: Combine the results from Step 2 and Step 3. The derivative of g(t) is the sum of the derivatives of its terms, so g'(t) = 2 + 1 / (t ln(5)).

Step 5: Simplify the expression if necessary. The final derivative is g'(t) = 2 + 1 / (t ln(5)). This is the simplified form of the derivative.

4:50m

Watch next

Master Derivatives of General Exponential Functions with a bite sized video explanation from Patrick

Start learning

Derivatives of Exponential & Logarithmic Functions practice set

Problem sets built by lead tutors

Expert video explanations

Go to Practice