Question

In Exercises \(1-8,\) use the given substitution and the Chain Rule to find \(d y / d x\) $$y=\sin (7-5 x), u=7-5 x$$

Short answer

The derivative \( \diff{y}{x} = -5 \cos(7 - 5x) \)

Step-by-step solution

Differentiate Outer Function

Firstly, differentiate the outer function with respect to \( u \). Since the outer function is \( y = \sin(u) \), its derivative with respect to \( u \) is \( \diff{y}{u} = \cos(u) \)

Differentiate Inner Function

Then, differentiate the inner function with respect to \( x \). The inner function is \( u = 7 - 5x \), its derivative with respect to \( x \) is \( \diff{u}{x} = -5 \)

Apply Chain Rule

Now, apply the chain rule. The chain rule is simply the multiplication of the derivative of the outer function by the derivative of the inner function. Giving, \( \diff{y}{x} = \diff{y}{u} * \diff{u}{x} = \cos(u) * -5 \)

Substitute Value of u Back

We initially performed a substitution for the function. Therefore, we should put the original value of \( u \) back into the equation. Here, \( u = 7 - 5x \), so the final derivative \( \diff{y}{x} = -5 \cos(7 - 5x) \)