Compute the derivative. (d)/(d t) log10(t+2^t)

Name: Problem 24, Chapter 3: Calculus: Early Transcendentals
Uploaded: 2021-07-20T01:26:50-08:00
Duration: 3 min 23 s
Channel: Ashley Boni
Description: Compute the derivative. (d)/(d t) log10(t+2^t)

Key Concepts

Chain Rule

The chain rule is fundamental for differentiating composite functions, and it states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. This rule is crucial when the argument of a function (like a logarithm) is itself another function.

Differentiation of Logarithmic Functions

Differentiating logarithmic functions involves applying specific rules that take into account the base of the logarithm. For logarithms with bases other than e, the derivative formula typically involves the natural logarithm of the base. This requires modifying the standard derivative of the natural logarithm to suit the function's base.

Derivative of Exponential Functions

When differentiating exponential functions, such as those where the independent variable is in the exponent, one uses the rule that the derivative of a^t (with a positive constant a) is a^t ln(a). This concept is important for functions where the exponent contains the variable being differentiated.

Change of Base in Logarithms

The change of base formula allows conversion from a logarithm of any base to one with base e, which is typically easier to differentiate. This concept is essential because it introduces a factor involving ln(a), where a is the original logarithm base, into the differentiation formula.

Transcript

00:01 So we want to find the derivative with respect to t of log base 10 of t plus 2 to the t.

00:09 So this is a little bit involved here, but our first step is we need to look at the outside function, which is the log.

00:18 So to find the derivative of a log, so remember we have this formula, if we want to find the derivative with respect to x of log base a of x, this is equal to 1 over x times the natural log of the base.

00:39 So we're going to use this idea here.

00:43 So we've got 1 over x is the thing inside parentheses.

00:51 So in our case, it's t plus 2 to the t.

00:56 And then we take ln of the base, which is 10 in our situation.

01:03 But we're not done yet.

01:04 We have an inside function as well.

01:06 It's not just an x in here or a t in here.

01:09 It's t plus two to the t.

01:11 So we need to take the derivative of the inside function as well.

01:15 So that's going to be our chain rule.

01:17 So chain rule is to take the derivative of t plus two to the t with respect to t, of course.

01:30 So let's figure out first what's our derivative.

01:33 So t plus two to the t.

01:37 If we take the derivative with respect to t, what is this equal to? well, the derivative of t is just one plus what's the derivative of two to the t? let's recall another formula...

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