Question

For the following exercises, compute \(d y / d x\) by differentiating \(\ln y\). $$ y=e^{\sin x} $$

Short answer

\( \frac{dy}{dx} = e^{\sin x} \cos x \)

Step-by-step solution

Apply Natural Logarithm

Start by applying the natural logarithm to both sides of the equation to make differentiation easier. Thus we have:\[ \ln y = \ln(e^{\sin x}) \]Using the property of logarithms, this simplifies to:\[ \ln y = \sin x \]

Differentiate Both Sides

Now differentiate both sides with respect to \( x \). On the left side, use the chain rule for differentiation:\[\frac{d}{dx}(\ln y) = \frac{1}{y} \cdot \frac{dy}{dx}\]On the right side, differentiate \( \sin x \) to get:\[\frac{d}{dx}(\sin x) = \cos x\]

Set Derivatives Equal

Now that each side has been differentiated, set them equal to each other:\[ \frac{1}{y} \cdot \frac{dy}{dx} = \cos x\]

Solve for \( \frac{dy}{dx} \)

To solve for \( \frac{dy}{dx} \), multiply both sides by \( y \):\[\frac{dy}{dx} = y \cos x\]Substitute back \( y = e^{\sin x} \) into the equation:\[\frac{dy}{dx} = e^{\sin x} \cos x\]

Key concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e\), where \( e \approx 2.71828 \). It is the inverse function of the natural exponential function. This function is commonly used in calculus, especially for its properties that make differentiation and integration more manageable.

When you apply natural logs to both sides of an equation, you simplify the differentiation process. For example, if \( y = e^{\sin x} \), taking the natural log gives us \( \ln y = \sin x \). This simplification exploits the logarithmic property that \( \ln(a^b) = b \cdot \ln a \), and since \( \ln e = 1 \), it reduces directly to the exponent \( b \).

Chain rule

The chain rule is a fundamental differentiation method used when dealing with composite functions. A composite function is essentially a function within another function. The chain rule states that to differentiate a composite function \( f(g(x)) \), you multiply the derivative of the outer function \( f \) evaluated at the inner function \( g(x) \) by the derivative of the inner function \( g \).

In our example, we dealt with \( \ln y = \sin x \), and needed to differentiate \( \ln y \) with respect to \( x \). This began with \( \frac{d}{dx}(\ln y) = \frac{1}{y} \cdot \frac{dy}{dx} \), an application of the chain rule where the "outer" is \( \ln y \) and the "inner" is \( y \).

Exponential function

The exponential function is denoted as \( e^x \), and involves the constant \( e \), an irrational number approximately equal to 2.71828. Exponential functions, noted for their unique rate of growth, play a crucial role in mathematics and its applications.

During differentiation, the exponential function holds a unique property: the derivative of \( e^{u(x)} \), where \( u(x) \) is any function of \( x \), is simply \( e^{u(x)} \cdot u'(x) \). This is particularly straightforward when paired with the chain rule.

In the original problem, once the expression \( y = e^{\sin x} \) is differentiated using the chain rule, it yields \( \cos x \), since the derivative of \( \sin x \) is \( \cos x \). The result \( \frac{dy}{dx} = e^{\sin x} \cos x \) highlights how this property simplifies differentiation.

Trigonometric function

Trigonometric functions like sine, cosine, and tangent are vital in calculus and beyond. They relate the angles of a triangle to the ratios of its sides and extend to periodic phenomena like waves.

When differentiating trigonometric functions, specific rules apply. For instance, the derivative of \( \sin x \) is \( \cos x \), which we used in our exercise.

Understanding how these functions transform under differentiation is crucial. In our exercise, differentiating \( \sin x \) directly led us to \( \cos x \). Knowing how each of these functions behaves makes calculus, especially when combined with other functions like exponential and logarithmic functions, more intuitive.