Find the derivative of the given function.h(x)=4(x+3x)54h\left(x\right)=4^{\left(\sqrt{x}+3x\right)^{\frac54}}

<math alttext="five fourths times open paren the square root of x plus 3 x close paren times open paren 1 over 2 the square root of x plus 3 close paren times 4 raised to the exponent open paren the square root of x plus 3 x close paren raised to the five fourths power end exponent" data-value="\frac54\left(\sqrt{x}+3x\right)\left(\frac{1}{2\sqrt{x}}+3\right)\cdot4^{\left(\sqrt{x}+3x\right)^{\frac54}}"> 5 4 ( x + 3 x ) ( 1 2 x + 3 ) ⋅ 4 ( x + 3 x ) 5 4 \frac54\left(\sqrt{x}+3x\right)\left(\frac{1}{2\sqrt{x}}+3\right)\cdot4^{\left(\sqrt{x}+3x\right)^{\frac54}}

Find the derivative of the given function. h ( x ) = 4 ( x + 3 x ) 5 4 h\left(x\right)=4^{\left(\sqrt{x}+3x\right)^{\frac54}}

Here, we wanna find the derivative of the function h of x equals 4 to the power of the square root of x plus 3 x, and everything in that exponent is raised to the power of 5 4ths. Now in order to find this derivative, we're going to have to apply the chain rule multiple times. So let's go ahead and get started. Now we wanna find our derivative here, h prime of x. So starting with our outermost function, we have this exponential function, 4 raised to the power of all of this stuff. Then as we work our way inside, in that exponent, we have all of this stuff being raised to the power of 5 4ths. Then inside of those parentheses, this is our innermost function, we have the square root of x plus 3 x. So we're starting from the outside, working our way in using the chain rule. So, again, that outermost function, 4 raised to the power of all of this stuff. I know that the derivative of an exponential function is going to be that base raised to the power of everything that is there. So here it is the square root of x plus 3x raised to the power of 5 4ths times the natural log of that base, in this case, 4. Now working our way inside here, we now want to find the derivative of the stuff raised to the power of 5 4ths, and we're multiplying by it. This is what our chain rule tells us. So we're going to apply the power rule here. We're going to pull that 5 4ths out in front of those parentheses. So that is 5 4ths times root x plus 3x, and that gets raised to the power of 5 4ths minus 1 using that Power Rule. 5 4ths minus 1 gives us 1 4th. Then finally, one more function working our way inside. We wanna multiply by the derivative of root x plus 3x. So one final derivative to take here, the square root of x. Remember that you can think of this as being x to the power of 1 half. So its derivative is 1 half x to the power of negative one half using the power rule. Then the derivative of 3x is just 3. Now we can rewrite this slightly. There is quite a bit going on. Typically, we want to have any constants out front. So I'm going to go ahead and pull that 5 fourths out front. So this is 5 over 4 times 4 to the power of the square root of x plus 3x raised to the power of 5 4ths times the natural log of 4. And then this function here, so times the square root of x plus 3x raised to the power of 1 4th. I'm just going to keep that as a fraction for now in that power. And then multiply it here by 1 over 2 root x plus 3. I'm just going to keep this factored as is. And this is my as is, and this is my full derivative here, h prime of x, having used the chain rule to work with all of these functions that are all composed with each other. If you have any questions here, feel free to let us know. I know that these chain rule problems can sometimes get a bit tricky. I'll see you in the next one.

Find the derivative of the given function. h ( x ) = 4 ( x + 3 x ) 5 4 h\left(x\right)=4^{\left(\sqrt{x}+3x\right)^{\frac54}}

<math alttext="five fourths times open paren the square root of x plus 3 x close paren times open paren 1 over 2 the square root of x plus 3 close paren times 4 raised to the exponent open paren the square root of x plus 3 x close paren raised to the five fourths power end exponent" data-value="\frac54\left(\sqrt{x}+3x\right)\left(\frac{1}{2\sqrt{x}}+3\right)\cdot4^{\left(\sqrt{x}+3x\right)^{\frac54}}"> 5 4 ( x + 3 x ) ( 1 2 x + 3 ) ⋅ 4 ( x + 3 x ) 5 4 \frac54\left(\sqrt{x}+3x\right)\left(\frac{1}{2\sqrt{x}}+3\right)\cdot4^{\left(\sqrt{x}+3x\right)^{\frac54}}

5 ln ⁡ ( 4 ) 4 ( x + 3 x ) ⋅ 4 ( x + 3 x ) 5 4 \frac{5\ln\left(4\right)}{4}\left(\sqrt{x}+3x\right)\cdot4^{\left(\sqrt{x}+3x\right)^{\frac54}}

<math alttext="five fourths times 4 raised to the exponent open paren the square root of x plus 3 x close paren raised to the five fourths power end exponent times l n 4 times open paren the square root of x plus 3 x close paren raised to the one fourth power times open paren 1 over 2 the square root of x plus 3 close paren" data-value="\frac54\cdot4^{\left(\sqrt{x}+3x\right)^{\frac54}}\cdot\ln4\cdot\left(\sqrt{x}+3x\right)^{\frac14}\cdot\left(\frac{1}{2\sqrt{x}}+3\right)"> 5 4 ⋅ 4 ( x + 3 x ) 5 4 ⋅ ln ⁡ 4 ⋅ ( x + 3 x ) 1 4 ⋅ ( 1 2 x + 3 ) \frac54\cdot4^{\left(\sqrt{x}+3x\right)^{\frac54}}\cdot\ln4\cdot\left(\sqrt{x}+3x\right)^{\frac14}\cdot\left(\frac{1}{2\sqrt{x}}+3\right)

( x + 3 x ) 5 4 ⋅ 4 ( x + 3 x ) 1 4 \left(\sqrt{x}+3x\right)^{\frac54}\cdot4^{\left(\sqrt{x}+3x\right)^{\frac14}}

4. Derivatives of Exponential & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Business Calculus

4. Derivatives of Exponential & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Struggling with Business Calculus?

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Multiple Choice

Find the derivative of the given function.
$h (x) = 4^{{(\sqrt{x} + 3 x)}^{\frac{5}{4}}}$

$\frac{5}{4} (\sqrt{x} + 3 x) (\frac{1}{2 \sqrt{x}} + 3) \cdot 4^{{(\sqrt{x} + 3 x)}^{\frac{5}{4}}}$

$\frac{5 \ln (4)}{4} (\sqrt{x} + 3 x) \cdot 4^{{(\sqrt{x} + 3 x)}^{\frac{5}{4}}}$

$\frac{5}{4} \cdot 4^{{(\sqrt{x} + 3 x)}^{\frac{5}{4}}} \cdot \ln 4 \cdot {(\sqrt{x} + 3 x)}^{\frac{1}{4}} \cdot (\frac{1}{2 \sqrt{x}} + 3)$

${(\sqrt{x} + 3 x)}^{\frac{5}{4}} \cdot 4^{{(\sqrt{x} + 3 x)}^{\frac{1}{4}}}$

Verified step by step guidance

Step 1: Identify the function to differentiate. The given function is h(x) = 4(√x + 3x)^(5/4). This is a composite function, so we will need to use the chain rule to differentiate it.

Step 2: Apply the chain rule. The chain rule states that if you have a composite function f(g(x)), its derivative is f'(g(x)) * g'(x). Here, the outer function is 4u^(5/4), where u = (√x + 3x).

Step 3: Differentiate the outer function. The derivative of 4u^(5/4) with respect to u is (5/4) * 4 * u^(1/4). Substituting u = (√x + 3x), this becomes (5/4) * 4 * (√x + 3x)^(1/4).

Step 4: Differentiate the inner function u = (√x + 3x). The derivative of √x is (1/2√x), and the derivative of 3x is 3. So, the derivative of u is (1/2√x) + 3.

Step 5: Combine the results. Multiply the derivative of the outer function by the derivative of the inner function. This gives [(5/4) * 4 * (√x + 3x)^(1/4)] * [(1/2√x) + 3]. Simplify this expression to get the final derivative.

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