Chain Rule Practice Problems

Problem: Chain Rule #1

Find the derivative of:

$\Large{y = (3x^2+5x^3)^2}$

Solution

Step 1:

We notice that the given function is a composite function, or a function within another function. To find the derivative of composite functions, we will use the chain rule. We can see that the inner function is $3x^2+5x^3$ and the outer function is the power of 2.

The chain rule states:

\Large{\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)}

Step 2:

Applying the chain rule, we have:

\Large{\frac{d}{dx}(3x^2+5x^3)^2 = 2(3x^2+5x^3)^{2-1} \cdot \frac{d}{dx}(3x^2+5x^3)}

Step 3:

Simplifying, we have:

\Large{ = 2(3x^2+5x^3)(6x+15x^2)}

Then, multiplying out everything, we end up with:

\Large{ = 150x^5+150x^4+36x^3}

Thus, this is the final answer for the derivative. So:

\Large{\frac{dy}{dx} = 150x^5+150x^4+36x^3}

Solution Complete!

Solution

Step 1:

The chain rule states:

\large{\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)}

Step 2:

Applying the chain rule, we have:

\large{\frac{d}{dx}(3x^2+5x^3)^2}

\small{ = 2(3x^2+5x^3)^{2-1} \cdot \frac{d}{dx}(3x^2+5x^3)}

Step 3:

Simplifying, we have:

\large{ = 2(3x^2+5x^3)(6x+15x^2)}

Then, multiplying out everything, we end up with:

\large{ = 150x^5+150x^4+36x^3}

Thus, this is the final answer for the derivative. So:

\large{\frac{dy}{dx} = 150x^5+150x^4+36x^3}

Solution Complete!

Problem: Chain Rule #2

Find the derivative of:

$\Large{f(x) = \sin(4x^2+1)}$

Problem: Chain Rule #3

Find the derivative of:

$\Large{h(x) = \sqrt{e^{2x}}}$

Problem: Chain Rule #4

Find the derivative of:

$\Large{f(x) = e^{1-x^2}}$

Problem: Chain Rule #5

Find the derivative of:

$\Large{g(x) = \cos(\ln(x))}$

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