Question

Compute the derivative of the given function. $$h(t)=e^{3 t^{2}+t-1}$$

Short answer

The derivative is \( h'(t) = (6t + 1) e^{3t^2 + t - 1} \).

Step-by-step solution

Identify the Function and Exponent

The function to differentiate is given as \( h(t) = e^{3t^2 + t - 1} \). Here, \(3t^2 + t - 1\) is the exponent of the natural exponential function \( e \).

Apply the Chain Rule

To differentiate \( h(t) \), we apply the chain rule. The chain rule states that the derivative of \( e^{u(t)} \) is \( e^{u(t)} \cdot u'(t) \). Here, \( u(t) = 3t^2 + t - 1 \).

Differentiate the Exponent

Find the derivative of \( u(t) = 3t^2 + t - 1 \). The derivative is obtained by differentiating term-by-term: \( \frac{d}{dt}(3t^2) = 6t \), \( \frac{d}{dt}(t) = 1 \), and \( \frac{d}{dt}(-1) = 0 \). Hence, \( u'(t) = 6t + 1 \).

Combine Results to Find the Derivative

Using the chain rule, the derivative of \( h(t) = e^{3t^2 + t - 1} \) is \( h'(t) = e^{3t^2 + t - 1} \cdot (6t + 1) \). Therefore, \( h'(t) = (6t + 1) e^{3t^2 + t - 1} \).

Key concepts

chain rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. You often need it when you're dealing with a function inside another function, like in this exercise where the function is of the form \( e^{u(t)} \).To apply the chain rule, you follow these steps:

• Identify the inner function, \( u(t) \), which in this circumstance is \( 3t^2 + t - 1 \).
• Differentiate \( u(t) \) to find \( u'(t) \).
• Multiply the derivative of the outer function with respect to \( u(t) \) by \( u'(t) \).

This means that for the given function \( h(t) = e^{u(t)} \), its derivative \( h'(t) \) is \( e^{u(t)} \cdot u'(t) \). It's like peeling back layers of an onion, dealing with one function at a time.

exponential function

An exponential function is characterized by having a constant base raised to a variable exponent. Here, the base is the mathematical constant \( e \) (approximately 2.718), making it a 'natural exponential function'.Natural exponential functions, like \( e^x \), have a unique property: they are their own derivatives. This property simplifies differentiation because when differentiating \( e^{u(t)} \):

• The base \( e \) stays the same after differentiation, simplifying expressions.
• The exponent function itself, \( u(t) \), has to be differentiated separately.

In this exercise, \( e \) remains constant, and you primarily work with the exponent to apply the chain rule effectively.

differentiation

Differentiation is the process of finding how a function changes as its input changes. It's a central operation in calculus and is used to determine rates of change and slopes of curves.To differentiate a function means to find its derivative. The derivative provides important information about the function, such as:

• Where it is increasing or decreasing.
• The location of maximum or minimum values.
• The slope of the function at any point.

In this specific problem, you differentiate \( h(t) \) to understand how it changes with respect to \( t \). Using methods like the chain rule helps to break down complex functions into simpler parts, making differentiation straightforward.

exponent

An exponent in mathematics refers to the power to which a number or expression is raised. It essentially represents repeated multiplication.In this exercise, the exponent \( 3t^2 + t - 1 \) is of a special type: it's a polynomial function. To deal with exponents like this in differentiation:

• Understand what the exponent represents (in this case, an expression of \( t \)).
• Differentiation involves writing an expression for the rate at which this exponent changes.
• Apply the derivative to each term separately: for example, the derivative of \( 3t^2 \) is \( 6t \).

This understanding helps compute the overall derivative accurately, especially when using rules like the chain rule.