Find the derivative of the given function.g(x)=ex+lnx5g(x)=e^{x}+\ln x^5g(x)=ex+lnx5

e x + 5 x e^{x}+\frac{5}{x} e x + x 5

Find the derivative of the given function. g ( x ) = e x + ln x 5 g(x)=e^{x}+\ln x^5 g ( x ) = e x + ln x 5

we're asked to find the derivative of the function g of x equals e to the x plus the natural log of x to the power of 5. Now we can actually find this derivative using the chain rule or without using the chain rule, and I'm gonna show you both methods here. Now looking at this first method without using the chain rule, we're actually gonna use properties of logs in order to find this. So no chain rule here. This is our first method to find our derivative, g prime of x. Now since I have 2 functions being added together, I know they can just separately add their derivatives. I know the derivative of e to the x is just e to the x, it's itself. Then if I look at this natural log of x to the power of 5, using my properties of logs, I can pull that 5 out front. That makes this 5 times the natural log of x. Now I know that if I take a constant and I multiply it by something, I just need to multiply that constant by the derivative in order to actually get that derivative. So this is 5 times the derivative of the natural log of x, which I know is just one over x. I can rewrite this a little bit as e to the x plus 5 over x, and that's how we get the derivative here with no chain rule. Now if instead you wanted to use the chain rule, so if we use the chain rule over here, in order to find this derivative at g prime of x, so we're not going to use any log properties here. Again, we know that the derivative of e to the x is just e to the x. But applying the chain rule here, we know that the derivative of the natural log is going to be one over whatever is in that natural log. So that makes this a natural log is going to be 1 over whatever is in that natural log. So that makes this one over x to the power of 5. But then applying the chain rule, we want to multiply this by the derivative of x to the power of 5. Using the power rule here, this is going to be 5 times x to the power of 4. Now looking at this, x to the power of 4 on top is gonna cancel with 4 of those on the bottom, so I'm just left with x down there. That makes this e to the x plus 5 over x, which is the exact same thing we got when applying our log properties. Both are acceptable methods in order to get this derivative. Let us know if you have questions, and let's keep going.

Find the derivative of the given function. g ( x ) = e x + ln ⁡ x 5 g(x)=e^{x}+\ln x^5 g ( x ) = e x + ln x 5

e x + 5 x e^{x}+\frac{5}{x} e x + x 5

5 e x x \frac{5e^{x}}{x} x 5 e x

e x + 5 ln ⁡ x e^{x}+5\ln x e x + 5 ln x

e x + 5 x \frac{e^{x}+5}{x} x e x + 5

4. Derivatives of Exponential & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Business Calculus

4. Derivatives of Exponential & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

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Multiple Choice

Find the derivative of the given function.
$g(x)=e^{x}+\ln x^5$

$\frac{5e^{x}}{x}$

$e^{x}+5\ln x$

$\frac{e^{x}+5}{x}$

$e^{x}+\frac{5}{x}$

Verified step by step guidance

Step 1: Identify the function to differentiate. The given function is g(x) = e^x + ln(x^5).

Step 2: Apply the derivative rules. For the first term, e^x, the derivative is simply e^x because the derivative of e^x is itself.

Step 3: For the second term, ln(x^5), use the logarithmic property ln(a^b) = b * ln(a) to rewrite it as 5 * ln(x). Then, differentiate 5 * ln(x) using the chain rule. The derivative of ln(x) is 1/x, so the derivative of 5 * ln(x) is 5/x.

Step 4: Combine the results from Step 2 and Step 3. The derivative of g(x) is the sum of the derivatives of its terms: g'(x) = e^x + 5/x.

Step 5: Simplify the expression if necessary. The final derivative is g'(x) = e^x + 5/x.

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