Derivative of e^sinx

We now understand the derivative of e^sinx. It is commonly represented as d/dx (e^sinx) or (e^sinx)', and its value is e^sinx * cosx.

The function e^sinx has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Exponential Function: (e^x), where e is the base of natural logarithms.

Chain Rule: Rule for differentiating e^sinx because it is a composition of functions.

Trigonometric Functions: sin(x) and cos(x) are used in the derivative.

The derivative of e^sinx can be denoted as d/dx (e^sinx) or (e^sinx)'.

The formula we use to differentiate e^sinx is: d/dx (e^sinx) = e^sinx * cosx

The formula applies to all x, as the exponential function and sine function are continuous everywhere.

We can derive the derivative of e^sinx using proofs. To show this, we will use the chain rule along with the properties of exponential and trigonometric functions.

There are several methods we use to prove this, such as:

1. Using Chain Rule
2. Using Product Rule

We will now demonstrate that the differentiation of e^sinx results in e^sinx * cosx using the above-mentioned methods:

Using Chain Rule To prove the differentiation of e^sinx using the chain rule, We use the formula: Let u = sinx, then e^u = e^sinx

By the chain rule: d/dx [e^u] = e^u * du/dx u = sinx, so du/dx = cosx

Therefore, d/dx (e^sinx) = e^sinx * cosx Using Product Rule We will now prove the derivative of e^sinx using the product rule.

The step-by-step process is demonstrated below:

Here, we use the formula, Let y = e^sinx y = e^u, where u = sinx

The product rule formula: d/dx [u.v] = u'.v + u.v'

Since v = e^sinx is not a product, we use the chain rule as shown earlier.

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.

o understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes.

Higher-order derivatives make it easier to understand functions like e^sinx. For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x), is the result of the second derivative, and this pattern continues.

For the nth Derivative of e^sinx, we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change (continuing for higher-order derivatives).

When x is π/2, the derivative is e^sin(π/2) * 0 = 0 because cos(π/2) = 0. When x is 0, the derivative of e^sinx = e^0 * 1 = 1.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.

• Exponential Function: A mathematical function of the form e^x, where e is the base of natural logarithms.

• Chain Rule: A rule in calculus for differentiating the compositions of functions.

• Sine Function: A trigonometric function representing the y-coordinate of a point on the unit circle.

• Product Rule: A rule for differentiating the product of two functions, stating d/dx [u.v] = u'.v + u.v'.