Math / Calculus

QuestionFind the derivative of the function. $\begin{array}{l} f(x)=4 e^{x}-x^{2} \\ f^{\prime}(x)=\square \end{array}$

Studdy Solution

STEP 1

What is this asking? We need to find the derivative of a function that has an exponential term and a power term. Watch out! Don't forget the rules for differentiating exponentials and powers – they're slightly different!

STEP 2

1. Differentiate the exponential term.
2. Differentiate the power term.
3. Combine the results.

STEP 3

The first term in our function $f(x)$ is $4e^x$ .
When we differentiate $e^x$ , we get $e^x$ back!
It's like magic!
Since the derivative of $e^x$ is $e^x$ , the derivative of $4 \cdot e^x$ is $4 \cdot e^x$ .

STEP 4

Now, let's look at the second term, $-x^2$ .
Remember the power rule: the derivative of $x^n$ is $n \cdot x^{n-1}$ .

STEP 5

Applying the power rule to $-x^2$ , we bring the exponent (which is $2$ ) down in front and reduce the exponent by $1$ .
So, the derivative of $-x^2$ becomes $-2 \cdot x^{2-1} = -2x$ .

STEP 6

We've found the derivatives of both parts of our function.
The derivative of $4e^x$ is $4e^x$ , and the derivative of $-x^2$ is $-2x$ .

STEP 7

Putting it all together, the derivative of the entire function $f(x) = 4e^x - x^2$ is $f'(x) = 4e^x - 2x$ .

STEP 8

$f'(x) = 4e^x - 2x$

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QuestionFind the derivative of the function. $\begin{array}{l} f(x)=4 e^{x}-x^{2} \\ f^{\prime}(x)=\square \end{array}$

Studdy Solution

STEP 1

What is this asking? We need to find the derivative of a function that has an exponential term and a power term. Watch out! Don't forget the rules for differentiating exponentials and powers – they're slightly different!

STEP 2

1. Differentiate the exponential term.
2. Differentiate the power term.
3. Combine the results.

STEP 3

The first term in our function $f(x)$ is $4e^x$ .
When we differentiate $e^x$ , we get $e^x$ back!
It's like magic!
Since the derivative of $e^x$ is $e^x$ , the derivative of $4 \cdot e^x$ is $4 \cdot e^x$ .

STEP 4

Now, let's look at the second term, $-x^2$ .
Remember the power rule: the derivative of $x^n$ is $n \cdot x^{n-1}$ .

STEP 5

Applying the power rule to $-x^2$ , we bring the exponent (which is $2$ ) down in front and reduce the exponent by $1$ .
So, the derivative of $-x^2$ becomes $-2 \cdot x^{2-1} = -2x$ .

STEP 6

We've found the derivatives of both parts of our function.
The derivative of $4e^x$ is $4e^x$ , and the derivative of $-x^2$ is $-2x$ .

STEP 7

Putting it all together, the derivative of the entire function $f(x) = 4e^x - x^2$ is $f'(x) = 4e^x - 2x$ .

STEP 8

$f'(x) = 4e^x - 2x$

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QuestionFind the derivative of the function. f(x)=4ex−x2f′(x)=□\begin{array}{l} f(x)=4 e^{x}-x^{2} \\ f^{\prime}(x)=\square \end{array}f(x)=4ex−x2f′(x)=□​

Studdy Solution

STEP 1

What is this asking? We need to find the *derivative* of a function that has an exponential term and a power term. Watch out! Don't forget the rules for differentiating exponentials and powers – they're slightly different!

STEP 2

1. Differentiate the exponential term.2. Differentiate the power term.3. Combine the results.

STEP 3

The first term in our function f(x)f(x)f(x) is 4ex4e^x4ex.When we **differentiate** exe^xex, we get exe^xex back!It's like magic!Since the derivative of exe^xex is exe^xex, the derivative of 4⋅ex4 \cdot e^x4⋅ex is 4⋅ex4 \cdot e^x4⋅ex.

STEP 4

Now, let's look at the second term, −x2-x^2−x2.Remember the **power rule**: the derivative of xnx^nxn is n⋅xn−1n \cdot x^{n-1}n⋅xn−1.

STEP 5

Applying the power rule to −x2-x^2−x2, we bring the **exponent** (which is 222) down in front and **reduce the exponent** by 111.So, the derivative of −x2-x^2−x2 becomes −2⋅x2−1=−2x-2 \cdot x^{2-1} = -2x−2⋅x2−1=−2x.

STEP 6

We've found the derivatives of both parts of our function.The derivative of 4ex4e^x4ex is 4ex4e^x4ex, and the derivative of −x2-x^2−x2 is −2x-2x−2x.

STEP 7

Putting it all together, the **derivative** of the entire function f(x)=4ex−x2f(x) = 4e^x - x^2f(x)=4ex−x2 is f′(x)=4ex−2xf'(x) = 4e^x - 2xf′(x)=4ex−2x.

STEP 8

f′(x)=4ex−2x f'(x) = 4e^x - 2x f′(x)=4ex−2x

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QuestionFind the derivative of the function. $\begin{array}{l} f(x)=4 e^{x}-x^{2} \\ f^{\prime}(x)=\square \end{array}$

What is this asking? We need to find the derivative of a function that has an exponential term and a power term. Watch out! Don't forget the rules for differentiating exponentials and powers – they're slightly different!

1. Differentiate the exponential term.
2. Differentiate the power term.
3. Combine the results.

The first term in our function $f(x)$ is $4e^x$ .
When we differentiate $e^x$ , we get $e^x$ back!
It's like magic!
Since the derivative of $e^x$ is $e^x$ , the derivative of $4 \cdot e^x$ is $4 \cdot e^x$ .

Now, let's look at the second term, $-x^2$ .
Remember the power rule: the derivative of $x^n$ is $n \cdot x^{n-1}$ .

Applying the power rule to $-x^2$ , we bring the exponent (which is $2$ ) down in front and reduce the exponent by $1$ .
So, the derivative of $-x^2$ becomes $-2 \cdot x^{2-1} = -2x$ .

We've found the derivatives of both parts of our function.
The derivative of $4e^x$ is $4e^x$ , and the derivative of $-x^2$ is $-2x$ .

Putting it all together, the derivative of the entire function $f(x) = 4e^x - x^2$ is $f'(x) = 4e^x - 2x$ .

$f'(x) = 4e^x - 2x$