Find the derivative of the given function.f(x)=2x−5xf(x)=2^x-5^xf(x)=2x−5x

2 x ln ⁡ 2 − 5 x ln ⁡ 5 2^{x}\ln2-5^{x}\ln5 2 x ln 2 − 5 x ln 5

Find the derivative of the given function. f ( x ) = 2 x − 5 x f(x)=2^x-5^x f ( x ) = 2 x − 5 x

In this problem, we wanna find the derivative of the function f of x equals 2 to the x minus 5 to the x. Now we just learned that the derivative of a general exponential function, so ddx of some base b to the power of x, is going to be that b to the power of x times the natural log of that base b. Now since here I have one exponential function being subtracted from another, that means I can look at each of their derivatives individually and just subtract them using the difference rule for derivatives. So my derivative here, f prime of x, using this rule that we just learned over here focusing on that 2 to the x, this is going to be 2 to the x times the natural log of 2. Then I'm going to subtract the derivative of 5 to the x, which is going to be 5 to the x times the natural log of 5, again, just using that rule. So this is my full derivative here. Let me know if you have any questions, and let's keep going.

Find the derivative of the given function. f ( x ) = 2 x − 5 x f(x)=2^x-5^x f ( x ) = 2 x − 5 x

2 x ln ⁡ 2 − 5 x ln ⁡ 5 2^{x}\ln2-5^{x}\ln5 2 x ln 2 − 5 x ln 5

x 2 ln ⁡ 2 − x 5 ln ⁡ ⁡ 5 x^2\ln2-x^5\ln⁡5 x 2 ln 2 − x 5 ln ⁡5

2 x ln ⁡ x − 5 x ln ⁡ ⁡ x 2^{x}\ln x-5^{x}\ln⁡x 2 x ln x − 5 x ln ⁡ x

2 ln ⁡ ⁡ x − 5 ln ⁡ x 2\ln⁡x-5\ln x 2 ln ⁡ x − 5 ln x

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Calculus

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

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Multiple Choice

Find the derivative of the given function.
$f(x)=2^x-5^x$

$x^2\ln2-x^5\ln5$

$2^{x}\ln2-5^{x}\ln5$

$2^{x}\ln x-5^{x}\lnx$

$2\lnx-5\ln x$

Verified step by step guidance

Identify the function for which you need to find the derivative: $ f(x) = 2^x - 5^x $.

Recall the derivative rule for exponential functions: $ \frac{d}{dx} a^x = a^x \ln a $, where $ a $ is a constant.

Apply the derivative rule to each term separately. For $ 2^x $, the derivative is $ 2^x \ln 2 $. For $ 5^x $, the derivative is $ 5^x \ln 5 $.

Combine the derivatives of each term to find the derivative of the entire function: $ f'(x) = 2^x \ln 2 - 5^x \ln 5 $.

Verify the result by checking each step and ensuring the application of the derivative rule is correct for exponential functions.

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