Question

Logarithmic differentiation Use logarithmic differentiation to evaluate \(f^{\prime}(x)\). $$f(x)=\left(1+x^{2}\right)^{\sin x}$$

Short answer

Question: Find the derivative of the function $$f(x)=\left(1+x^{2}\right)^{\sin x}$$ using logarithmic differentiation. Answer: The derivative of the given function using logarithmic differentiation is $$f^{\prime}(x) = \left(1+x^{2}\right)^{\sin x} \cdot \left[(\cos x) \cdot \ln\left(1+x^{2}\right) + \sin x \cdot \frac{2x}{1+x^{2}}\right]$$.

Step-by-step solution

Rewrite the function using logarithm property

We'll make use of the logarithm property: $$\log_b a^c = c \log_b a$$ where a>0, a≠1 and b>0. Take the natural logarithm of both sides: $$\ln (f(x)) = \ln\left(\left(1+x^{2}\right)^{\sin x}\right)$$ Now rewrite the right side of the equation using the logarithm property: $$\ln (f(x)) = \sin x \cdot \ln\left(1 + x^{2}\right)$$

Differentiate both sides with respect to x

After differentiating both sides of the equation with respect to x, our task would be to find the derivative of $$\sin x \cdot \ln\left(1 + x^{2}\right)$$ using the product rule and simplify. Differentiate both sides, $$\frac{f^{\prime}(x)}{f(x)} = (\sin x)^{\prime} \cdot \ln\left(1+x^{2}\right) + \sin x \cdot\left(\ln\left(1+x^{2}\right)\right)^{\prime}$$

Apply product rule and chain rule to differentiate the right side

We'll apply the chain rule for differentiating $$\ln\left(1 + x^{2}\right)$$ and product rule for differentiating the whole right side: $$\frac{f^{\prime}(x)}{f(x)} = (\cos x) \cdot \ln\left(1+x^{2}\right) + \sin x \cdot\frac{2x}{1+x^{2}}$$

Solve for $$f^{\prime}(x)$$

Finally, we'll multiply both sides by $$f(x)$$ to obtain the derivative $$f^{\prime}(x)$$: $$f^{\prime}(x) = f(x) \cdot \left[(\cos x) \cdot \ln\left(1+x^{2}\right) + \sin x \cdot \frac{2x}{1+x^{2}}\right]$$ Now substitute the original function $$f(x) = \left(1+x^{2}\right)^{\sin x}$$: $$f^{\prime}(x) = \left(1+x^{2}\right)^{\sin x} \cdot \left[(\cos x) \cdot \ln\left(1+x^{2}\right) + \sin x \cdot \frac{2x}{1+x^{2}}\right]$$ So, the derivative of the given function using logarithmic differentiation is: $$f^{\prime}(x) = \left(1+x^{2}\right)^{\sin x} \cdot \left[(\cos x) \cdot \ln\left(1+x^{2}\right) + \sin x \cdot \frac{2x}{1+x^{2}}\right]$$

Key concepts

Derivative

In calculus, the derivative of a function measures how that function changes as its input changes. In simpler terms, it gives you the rate at which one quantity changes with respect to another. For the function \( f(x) = (1+x^2)^{\sin x} \), finding the derivative \( f'(x) \) involves determining how \( f(x) \) changes as \( x \) changes.
Derivatives can be applied to a variety of situations in science and engineering to predict trends and make informed decisions. The process of differentiation uses several rules and techniques to find the derivative of complex functions, such as using the product rule, chain rule, and logarithmic differentiation, which we will explore further below.

Product Rule

The product rule is a vital tool in calculus for finding the derivative of a product of two functions. When we have a function that is the product of two differentiable functions, say \( u(x) \) and \( v(x) \), the product rule states that the derivative \\( (uv)' = u'v + uv' \).
For our function, \( f(x) = \sin x \cdot \ln(1 + x^2) \), we need to differentiate using the product rule to account for both parts of the multiplication:

• Differentiating \( \sin x \) gives \( \cos x \).
• Differentiating \( \ln(1 + x^2) \) involves the chain rule as well, which we will discuss in the next section.

Combining these results using the product rule helps us develop a complete expression for the derivative.

Chain Rule

The chain rule is a fundamental theorem in calculus for finding the derivative of a composite function. When you have two functions inside each other, like \( g(f(x)) \), the chain rule states that the derivative is \( g'(f(x)) \, f'(x) \).
For example, in our function \( \ln(1 + x^2) \), we need to apply the chain rule because it is a log function with a composite expression inside it. Here's how it's applied:

• The outer function is \( \ln(u) \), whose derivative is \( \frac{1}{u} \).
• Where \( u = 1 + x^2 \), and the derivative of \( x^2 \) is \( 2x \).

Thus, the chain rule helps us find the inner function's derivative correctly and is essential for differentiating nested functions.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is an important logarithmic function with the base \( e \), where \( e \approx 2.71828 \). It's widely used in mathematics due to its unique properties, especially when it comes to simplifying differentiation and integration.
When we use the natural logarithm in differentiation, it can make complex derivatives easier to solve. This is called logarithmic differentiation. By taking the logarithm of both sides of an equation, you can turn products into sums, which are often easier to differentiate.
In the problem \( \ln((1 + x^2)^{\sin x}) \), we used the property \( \ln(a^b) = b \ln(a) \) to simplify the expression and make differentiation more straightforward. This is a crucial step in solving derivatives of functions involving power terms, especially when they consist of variable exponents.