Question

Given \(c=\sin (a-b), b=a e^{2 a},\) find \(d c / d a.\)

Short answer

The derivative is: cos(a - a e ^{2a}) * (1 - (a 2e ^ {2a} + e^{2a} ).

Step-by-step solution

Substitute Expression of b in c

Given that b = a e^{2a} , we need to substitute this into the c equation: c = sin(a - (a e^{2a})).

Simplify the Expression of c

After substitution, c simplifies to c = sin(a - a e^{2a} ).

Differentiate c with Respect to a

Use the chain rule for differentiation. Let f(a) = a - a e^{2a} , so c = sin(f(a)). The chain rule states that d c / d a = (d sin(f(a)) / d f(a)) * (d f(a) / d a). So d c / d a = cos(f(a)) * d f(a) / d a.

Differentiate f(a) with Respect to a

Differentiate f(a) = a - a e^{2a} with respect to a. Using the product rule and the chain rule, d f(a) / d a = 1 - (a * d ( e^{2a} ) / d a + e^{2a} ). This is equivalent to 1 - (a * 2 e^{2a} + e^{2a} ). Simplifying, we get d f(a) / d a = 1 - (a 2 e^{2a} + e^{2a} ) = 1 - ( a 2 e^{2a} + e^{2a} ).

Substitute f(a) and df(a)/da back into dc/da

Now that d f(a) / d a has been determined, substitute it back into the chain rule formula from Step 3: d c / d a = cos(a - a e^{2a}) * (1 - (a 2 e^{2a} + e^{2a} )).

Key concepts

chain rule

The chain rule is a fundamental tool for differentiating compositions of functions. If you have a function that is composed of multiple functions, you use the chain rule to find its derivative. For a function \(c = g(f(a))\), where \(g\) is a function of \(f\) and \(f\) is a function of \(a\), the chain rule states that the derivative of \(c\) with respect to \(a\) is the product of the derivative of \(g\) with respect to \(f\) and the derivative of \(f\) with respect to \(a\). Formally, this is written as:
\( \frac{dc}{da} = \frac{dg}{df} \cdot \frac{df}{da} \).
In our problem, we use the chain rule to break down the differentiation of the composite functions like \( \sin(f(a)) \).
It helps by tackling smaller parts first, making the problem easier. Remember, the chain rule is especially useful when functions are nested, like \(\sin\) within another function.

product rule

The product rule is used when differentiating a product of two functions. If you have two functions \(u(a)\) and \(v(a)\), their product differentiation is:
\( \frac{d}{da} [u(a) \cdot v(a)] = u'(a) \cdot v(a) + u(a) \cdot v'(a) \).
Notice that you differentiate each function once, while keeping the other function unchanged, and then sum the results.
In our problem, apply the product rule to differentiate \(a \cdot e^{2a}\) within the function \(f(a) = a - a e^{2a}\).
First differentiate \(a\), and then differentiate \(e^{2a}\), combining these results correctly.

implicit differentiation

Implicit differentiation involves differentiating a function that is not explicitly solved for one variable in terms of the other. Often, you use it when you have functions intertwined in a more complex expression.
While our specific exercise didn’t require implicit differentiation, it's a key concept to understand in calculus.
To perform implicit differentiation, differentiate each term with respect to the required variable, applying the chain rule where necessary, and then solve for the desired derivative.
It's common in problems involving curves and geometric shapes not easily isolated.

exponential function

An exponential function is one in the form \(e^x\), where \(e\) is the base of natural logarithms, approximately equal to 2.71828.
The exponential function has unique properties, such as being its own derivative.
When differentiating \(e^{2a}\), use the chain rule, giving \( \frac{d}{da} e^{2a} = 2e^{2a} \).
In our problem, this is used extensively for differentiating parts of the expression involving \(e^{2a}\).
Always keep in mind that when you see exponentials, derivatives might involve another chain rule application.

trigonometric functions

Trigonometric functions like sine and cosine are crucial in calculus.
The basic derivatives you must know are:

• \( \frac{d}{dx} \sin(x) = \cos(x) \)
• \( \frac{d}{dx} \cos(x) = -\sin(x) \)

For the exercise, the function \(c = \sin(a - a e^{2a})\) required using these derivatives. By combining the outer derivative of sine with the inner functions, you apply the chain rule.
Understanding how these trigonometric derivatives work allows you to handle more complex compositions easily.