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Chapter 3: Problem 15

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=\tan 5 x^{2}$$

Short Answer

Expert verified
Question: Find the derivative of the function \(y = \tan(5x^2)\) with respect to \(x\). Answer: The derivative of the function \(y=\tan(5x^2)\) with respect to \(x\) is \(\frac{dy}{dx} = 10x\sec^2(5x^2)\).

Step by step solution

01

Identify the inner and outer functions

In this case, the inner function is \(g(x) = 5x^2\) and the outer function is \(f(u)=\tan(u)\), where \(u = g(x)\).
02

Take the derivatives of inner and outer functions

Now, we need to find the derivatives of both functions. For the inner function: \(\frac{d}{dx}(5x^2) = 10x\) For the outer function, let's apply derivative to \(\tan(\cdot)\): \(\frac{d}{du}(\tan(u)) = \sec^2(u)\)
03

Apply the Chain Rule

According to the Chain Rule, we have \((f(g(x)))' = f'(g(x)) * g'(x)\). Substitute the derivatives we found in Step 2: $$ \frac{dy}{dx} = \frac{d}{dx}(\tan(5x^2)) = \sec^2(5x^2) * 10x $$
04

Final answer

So, the derivative of \(y=\tan(5x^2)\) with respect to \(x\) is: $$ \frac{dy}{dx} = 10x\sec^2(5x^2) $$

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