How do you differentiate #f(x)=sqrt(e^(5x^2+x+3) # using the chain rule?

Question

How do you differentiate #f(x)=sqrt(e^(5x^2+x+3)  # using the chain rule?

Ezra Adams · Accepted Answer

This will require the application of the chain rule--twice.

First, for #e^(5x^2 + x + 3)#

Let #y = e^u#, and #u = 5x^2 + x + 3#

#dy/dx = dy/(du) xx (du)/dx#

The derivative of #e^u# is #e^u#. The derivative of #5x^2 + x + 3# is #10x + 1#.

Hence,

#dy/dx = e^u xx 10x + 1#

#dy/dx = (10x + 1)e^(5x^2 + x + 3)#

Now for the second application of the chain rule.

Let #y = sqrt(u) = u^(1/2)# and #u = e^(5x^2 + x + 3)#

We already know the derivative of #u#. The derivative of #y#, by the power rule, is #1/2u^(-1/2) = 1/(2u^(1/2))#.

Hence,

#dy/dx = 1/(2u^(1/2)) xx (10x + 1)e^(5x^2 + x + 3)#

#dy/dx = ((10x + 1)e^(5x^2 + x + 3))/(2sqrt(e^(5x^2 + x + 3))#

#f'(x) = ((10x + 1)e^(5x^2 + x + 3))/(2sqrt(e^(5x^2 + x + 3))#

#f'(x) = ((10x + 1)e^(5x^2 + x + 3))/(2(e^(5x^2 + x + 3))^(1/2)#

By the quotient rule of exponents: #a^n/a^m = a^(n - m)#:

#f'(x) = ((10x + 1)sqrt(e^(5x^2 + x + 3)))/2#

Hopefully this helps!

Luke Alvarez · Answer

undefined To differentiate $ f(x) = \sqrt{e^{5x^2 + x + 3}} $ using the chain rule, follow these steps:

1. Let $ u = 5x^2 + x + 3 $.
2. Rewrite the function as $ f(x) = \sqrt{e^u} $.
3. Find the derivative of $ u $ with respect to $ x $, denoted as $ u' $.
   $ u' = \frac{d}{dx}(5x^2 + x + 3) = 10x + 1 $.
4. Apply the chain rule:
   $ \frac{d}{dx}\sqrt{e^u} = \frac{d}{du}\sqrt{e^u} \cdot \frac{du}{dx} $.
5. Derivative of $ \sqrt{e^u} $ with respect to $ u $ is $ \frac{1}{2\sqrt{e^u}} $.
6. Multiply by $ u' $ to get the final answer:
   $ f'(x) = \frac{1}{2\sqrt{e^u}} \cdot (10x + 1) $.
7. Substitute $ u = 5x^2 + x + 3 $ back into the equation:
   $ f'(x) = \frac{1}{2\sqrt{e^{5x^2 + x + 3}}} \cdot (10x + 1) $.

Isaiah Allen · Answer

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Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by theTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. FinallyTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivativeTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally,To differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative ofTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substituteTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of theTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substitute theTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of the innerTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substitute the originalTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner functionTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substitute the original functionTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.To differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substitute the original function andTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. TheTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substitute the original function and itsTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. The derivativeTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substitute the original function and its derivativeTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. The derivative ofTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substitute the original function and its derivative intoTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. The derivative of theTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substitute the original function and its derivative into theTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. The derivative of the squareTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substitute the original function and its derivative into the chainTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. The derivative of the square rootTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substitute the original function and its derivative into the chain ruleTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. The derivative of the square root functionTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substitute the original function and its derivative into the chain rule formulaTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. The derivative of the square root function isTo differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substitute the original function and its derivative into the chain rule formula.To differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. The derivative of the square root function is $To differentiate \( f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substitute the original function and its derivative into the chain rule formula.To differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. The derivative of the square root function is $ \To differentiate \( f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, differentiate the outer function with respect to its argument and multiply it by the derivative of the inner function with respect to $ x $. Finally, substitute the original function and its derivative into the chain rule formula.To differentiate $ f(x) = \sqrt{e^{5x^2+x+3}} $ using the chain rule, first, identify the outer function, which is the square root function, and the inner function, which is $ e^{5x^2+x+3} $. Then, apply the chain rule, which states that the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. The derivative of the square root function is $ \frac{1}{2\sqrt{x}} $, and the derivative of the inner function $ e^{5x^2+x+3} $ with respect to $ x $ is $ (10x + 1)e^{5x^2+x+3} $. Finally, combine these derivatives to obtain the derivative of the original function.