Question

Calculate the derivative of the following functions. $$y=e^{\tan t}$$

Short answer

Answer: The derivative of the given function with respect to t is y'(t) = e^{\tan(t)} \cdot \sec^2(t).

Step-by-step solution

Recall the chain rule

The chain rule states that if y = f(g(x)), then the derivative of y with respect to x is as follows: $$y'(x) = f'(g(x)) \cdot g'(x)$$ Here, our function is y = e^{\tan(t)}. Let us consider f(u) = e^u and g(t) = \tan(t), with y = f(g(t)).

Find the derivatives of f(u) and g(t)

We need to find the derivatives of f(u) and g(t) with respect to their respective variables: The derivative of f(u) = e^u with respect to u is: $$f'(u) = e^u$$ The derivative of g(t) = \tan(t) with respect to t is: $$g'(t) = \frac{d}{dt} (\tan(t)) = \sec^2(t)$$

Apply the chain rule

Now we apply the chain rule to find the derivative of y with respect to t: $$y'(t) = f'(g(t)) \cdot g'(t)$$ By substituting the values of f'(u) and g'(t) that we found in Step 2, we get: $$y'(t) = e^{\tan(t)} \cdot \sec^2(t)$$

Final answer

The derivative of the given function y=e^{\tan t} with respect to t is as follows: $$y'(t) = e^{\tan(t)} \cdot \sec^2(t)$$

Key concepts

Derivative of Exponential Functions

Understanding the derivative of exponential functions is crucial for many calculus problems. An exponential function can be expressed in the form of f(x) = e^x, where e is the base of the natural logarithm, approximately equal to 2.71828. The uniqueness of the exponential function lies in its property that the derivative remains the same as the original function. That is, if you take the derivative of e^x with respect to x, you will get e^x as the result.

In mathematical terms, this can be represented as:
\[ \frac{d}{dx}(e^x) = e^x \.
\] This property makes working with exponential functions much simpler because the function's growth rate is proportional to its current value, which is a unique characteristic of exponential growth.

Derivative of Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, and their derivatives follow specific rules. For instance, the derivative of tan(x), which is the ratio of the sine to the cosine of x, has its own derivative, which is not immediately obvious from the derivatives of sine and cosine.

The derivative of tan(x) with respect to x is given by the square of the secant of x: \[ \frac{d}{dx}(\tan(x)) = \sec^2(x) \.
\] Understanding this derivative rule is important because the tangent and secant functions are prevalent in advanced mathematics and physics. As tan(x) frequently appears in calculus, recognizing its derivative quickly makes the solving process much more efficient.

Applying the Chain Rule

The chain rule is a powerful tool in calculus, particularly when you're dealing with composite functions, where one function is nested inside another. When you have a composite function like y = f(g(x)), you can't simply take the derivative of the outer and inner functions separately; they're linked.

The chain rule tells us how to find the derivative of such a composite function, which is: \[ y'(x) = f'(g(x)) \cdot g'(x) \.
\] In practice, you compute the derivative of the outer function f(x) evaluated at g(x), and multiply it by the derivative of the inner function g(x) with respect to x. This chaining of derivatives allows us to navigate the relationship between the functions and is an essential skill for working through more complex differentiation problems. When you're applying the chain rule, careful substitution of terms and diligent simplification lead to the correct result.

Implicit Differentiation

Sometimes, functions come in forms that are not explicitly solved for one variable in terms of another, which is where implicit differentiation becomes a valuable technique. It allows us to find the derivative of a function with respect to a certain variable, even when that function is expressed in terms of both variables mingled together.

Implicit differentiation operates under the assumption that even if y is not isolated on one side of the equation, it is still a function of x, and therefore we can differentiate both sides of the equation with respect to x. Anytime you come across a derivative of y, we apply the chain rule because we're assuming y is a function of x. This strategy is used to find the derivative in situations where separating the variables would be too complex or impossible. It's a subtle yet potent part of calculus, easing the path to derivatives of many complex expressions.