Question

General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{d x}\left(1+x^{2}\right)^{\sin x}$$

Short answer

Question: Find the derivative of the function \(y = \left(1+x^{2}\right)^{\sin x}\) with respect to x. Answer: The derivative of the function is \(\frac{d}{dx} \left(1+x^{2}\right)^{\sin x} = \left(1+x^{2}\right)^{\sin x} \cdot \left[\cos{x} \cdot \ln{\left(1+x^{2}\right)} + \sin{x} \cdot \frac{2x}{1+x^{2}}\right]\).

Step-by-step solution

Rewrite the function using natural logarithm (ln)

Let y be the given function, then we have: $$y = \left(1+x^{2}\right)^{\sin x}$$ Take the natural logarithm of both sides: $$\ln{y} = \sin{x} \cdot \ln{\left(1+x^{2}\right)}$$

Differentiate both sides with respect to x

Using the chain rule on the left side, and product rule on the right side: $$\frac{1}{y} \cdot \frac{dy}{dx} = \cos{x} \cdot \ln{\left(1+x^{2}\right)} + \sin{x} \cdot \frac{2x}{1+x^{2}}$$

Solve for dy/dx

Now, we'll need to isolate the derivative \(\frac{dy}{dx}\) by multiplying both sides by y: $$\frac{dy}{dx} = y \cdot \left[\cos{x} \cdot \ln{\left(1+x^{2}\right)} + \sin{x} \cdot \frac{2x}{1+x^{2}}\right]$$

Substitute the original function for y

Replace y with \(\left(1+x^{2}\right)^{\sin x}\): $$\frac{dy}{dx} = \left(1+x^{2}\right)^{\sin x} \cdot \left[\cos{x} \cdot \ln{\left(1+x^{2}\right)} + \sin{x} \cdot \frac{2x}{1+x^{2}}\right]$$ So the derivative of the given function is: $$\frac{d}{dx} \left(1+x^{2}\right)^{\sin x} = \left(1+x^{2}\right)^{\sin x} \cdot \left[\cos{x} \cdot \ln{\left(1+x^{2}\right)} + \sin{x} \cdot \frac{2x}{1+x^{2}}\right]$$

Key concepts

Calculus

Calculus is a branch of mathematics that studies continuous change, and it is split into two subfields: differential calculus and integral calculus. Differential calculus is concerned with how functions change over time — that is, with finding the rate at which quantities change. This is where derivatives come into play. The derivative of a function at a particular point gives you the slope of the tangent line to the function's graph at that point, which is the rate of change of the function with respect to its variable.

In the given exercise, we are asked to find the derivative of a function that has both an exponential and a trigonometric component, challenging us to apply various calculus techniques to find the solution.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a logarithm to the base e, where e is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm of a number is the power to which e must be raised to produce that number. For example, if \(\ln(x) = y\), then \(e^y = x\).

Logarithmic differentiation is a technique where you take the natural logarithm of both sides of an equation before differentiating, as shown in the exercise. This technique is particularly useful when dealing with products, quotients, or powers of functions, which would be more complicated to differentiate using the standard rules.

Chain Rule

The chain rule is a formula for finding the derivative of a composite function. It states that if you have a function \(h(x) = f(g(x))\), then its derivative is \(h'(x) = f'(g(x)) \cdot g'(x)\). In simpler terms, you differentiate the outer function, leaving the inner function unchanged, and multiply by the derivative of the inner function.

In logarithmic differentiation, after taking the natural logarithm of both sides, we often end up with a composite function involving the natural logarithm function and the original function (the inner function). We apply the chain rule to differentiate it, as seen in the step-by-step solution of the exercise.

Product Rule

The product rule is a fundamental rule in calculus used to find the derivative of the product of two functions. It states that if you have two differentiable functions, \(u(x)\) and \(v(x)\), then the derivative of their product \(u(x) \cdot v(x)\) is \(u'(x) \cdot v(x) + u(x) \cdot v'(x)\).

This means that, to differentiate the product of two functions, you take the derivative of each function in turn and sum the two results after multiplying each by the other function. In the exercise at hand, the product rule is applied to the right side of the equation during the differentiation process after applying the natural logarithm.

Exponential Derivatives

Exponential derivatives refer to the derivatives of functions that involve an exponent, particularly when the exponent is a variable. For a general exponential function \(a^{x}\), where a is a constant, the derivative is \(a^{x} \ln(a)\) assuming \(a > 0\). But when you have a variable in the exponent, as \(x^y\) or in our exercise \( (1+x^2)^{\sin x} \), the differentiation gets more complex, requiring the use of logarithmic differentiation. By taking the natural logarithm of both sides, we can transform the power into a product, making it easier to apply the derivative rules.

Using logarithmic differentiation simplifies the process, allowing us to find the derivative of complex exponential functions, especially those involving variables or functions as exponents—precisely what was needed for the given problem.