Question

In Exercises, find the second derivative. $$ f(x)=5 e^{-x}-2 e^{-5 x} $$

Short answer

The second derivative of the function is \( f''(x) = 5e^{-x} -50e^{-5x} \).

Step-by-step solution

Differentiate \( f(x) \) to find \( f'(x) \)

Differentiate \( f(x) = 5e^{-x} - 2e^{-5x} \) with respect to \( x \), using the chain rule, which states that \( \frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x) \). For \( 5e^{-x} \), the derivative is \( -5e^{-x} \) and for \( -2e^{-5x} \), it is \( 10e^{-5x} \), hence \( f'(x) = -5e^{-x} + 10e^{-5x} \).

Derive \( f'(x) \) to obtain \( f''(x) \)

Differentiate \( f'(x) = -5e^{-x} + 10e^{-5x} \) with respect to \( x \), again using the chain rule. For \( -5e^{-x} \), the derivative is \( 5e^{-x} \) and for \( 10e^{-5x} \), it's \( -50e^{-5x} \), thus \( f''(x) = 5e^{-x} -50e^{-5x} \).

Key concepts

Chain Rule

Understanding the chain rule is essential for differentiating compositions of functions. In essence, this rule is a formula for computing the derivative of the composite of two or more functions. Imagine that we have two functions, one nested inside the other. If we want to find the derivative of this composite function, we first differentiate the outer function while keeping the inner function unchanged, and then multiply by the derivative of the inner function.

For the function in our exercise, which involves the exponential function with a negative exponent, the chain rule is applied by differentiating the exponential part as the 'outer function' and then multiplying by the derivative of the negative exponent, which acts as the 'inner function'. The process of applying the chain rule in our step-by-step solution ensures that each term is correctly differentiated, leading to the accurate calculation of the first derivative, and subsequently the second derivative when the process is repeated.

Exponential Functions Differentiation

The differentiation of exponential functions is another key concept showcased in the given exercise. Exponential functions are of the form e^(ax), where e is the base of the natural logarithm and a is a constant. One of the remarkable properties of e is that the derivative of e^(ax) is ae^(ax), making it straightforward to differentiate exponential functions. When we have a negative exponent like -x or -5x, the process is the same, but we need to multiply by the derivative of the exponent (applying the chain rule) which introduces a factor of -1 or -5, respectively. This property is applied in our solution to find the first derivative, where the derivative of 5e^(-x) is -5e^(-x) and the derivative of -2e^(-5x) is 10e^(-5x) due to the mentioned property in combination with the chain rule.

Higher-order Derivatives

Finally, higher-order derivatives refer to derivatives of a function taken multiple times. After finding the first derivative, we can proceed to find the second derivative, which is the derivative of the derivative, and so on. In the given exercise, we're asked to find the second derivative f''(x). This involves taking the derivative f'(x) and differentiating it once more. Repeatedly applying the differentiation rules, such as the chain rule and the rules for exponentials, allows us to systematically find the next level of derivatives.

Higher-order derivatives can provide additional information about the function's behavior, such as concavity and inflection points. The ability to compute these derivatives is not only essential for solving calculus problems but also for applications in physics, engineering, and economics, where the rate of change of a rate of change is often of interest.