Question

Производная f(x) = ch(x) + (5e)^(3x-x^2)

Answer 1

f'(x)=sh(x) +5e^(3x-x^2) * (3x-x^2)'= sh(x) +5e^(3x-x^2) * (3-2x)

Answer 2

To find the derivative of the function $f(x) = \cosh(x) + (5e)^{3x-x^2}$, we'll use the rules of differentiation. The derivative of $\cosh(x)$ is $\sinh(x)$, and the derivative of $(5e)^{3x-x^2}$ can be obtained using the chain rule. Let's proceed step by step:

Step 1: Find the derivative of $\cosh(x)$ $\frac{d}{dx}(\cosh(x)) = \sinh(x)$

Step 2: Find the derivative of $(5e)^{3x-x^2}$ using the chain rule. Let $u = 3x-x^2$ and $v = (5e)^u$. Then, we have: $\frac{dv}{dx} = \frac{dv}{du} \cdot \frac{du}{dx}$

To find $\frac{dv}{du}$, we'll use the power rule: $\frac{dv}{du} = \frac{d}{du}((5e)^u) = (5e)^u \cdot \ln(5e) = 5e^{3x-x^2} \cdot \ln(5e)$

To find $\frac{du}{dx}$, we'll use the power rule and the constant multiple rule: $\frac{du}{dx} = \frac{d}{dx}(3x-x^2) = 3 - 2x$

Now, combining these results, we get: $\frac{dv}{dx} = 5e^{3x-x^2} \cdot \ln(5e) \cdot (3 - 2x)$

Step 3: Combine the derivatives. The derivative of the function $f(x) = \cosh(x) + (5e)^{3x-x^2}$ is the sum of the derivatives from Step 1 and Step 2: $f'(x) = \sinh(x) + 5e^{3x-x^2} \cdot \ln(5e) \cdot (3 - 2x)$

So, the final derivative of the function $f(x) = \cosh(x) + (5e)^{3x-x^2}$ is $f'(x) = \sinh(x) + 5e^{3x-x^2} \cdot \ln(5e) \cdot (3 - 2x)$.