calculus infography
Avatar
CalculusPop
CalculusPop
Hey, let's solve some calculus with CalculusPop
 
Differentiation

Function of a function can be solved using the chain rule.

Here's a step-by-step guide on how to solve differentiation of a function of a function using the chain rule:

1. Identify the inner and outer functions:
Let y = f(g(x)), where f and g are functions. Identify g(x) as the inner function and f(u) as the outer function.

2. Differentiate the outer function with respect to the inner function:
Find the derivative of f(u) with respect to u, denoted as f'(u).

3. Differentiate the inner function with respect to x:
Find the derivative of g(x) with respect to x, denoted as g'(x).

4. Multiply the two derivatives together:
Multiply f'(u) and g'(x) together to get the derivative of the composite functions.

5. Replace u with g(x) in the final expression:
Substitute g(x) for u in the resulting expression to get the final derivative of y with respect to x.

1. Differentiate the function f(x) = (3x^2 + 1)^3.

Solution:

To differentiate a function of a function, we can use the chain rule. Let u = 3x^2 + 1. Then, f(x) = u^3.

Now, let's differentiate u with respect to x:

\frac{du}{dx} = 6x.

Now, applying the chain rule, we have:

\frac{df}{dx} = \frac{d(u^3)}{du} \cdot \frac{du}{dx}

\frac{df}{dx} = 3u^2 \cdot 6x

\frac{df}{dx} = 18x(3x^2 + 1)^2.

2. Find the derivative of g(x) = e^{4x^2}.

Solution:

To differentiate a function of a function, we can use the chain rule. Let u = 4x^2, then g(x) = e^u.
Now, let's differentiate u with respect to x:

\frac{du}{dx} = 8x.

Then, applying the chain rule, we have:

\frac{dg}{dx} = e^u \cdot 8x

\frac{dg}{dx} = 8x e^{4x^2}.

3. Differentiate the function h(x) = \sqrt{5x^3 + 2}.

Solution:

To differentiate a function of a function, we can use the chain rule. Let u = 5x^3 + 2, then h(x) = \sqrt{u}.

Now, let's differentiate u with respect to x:

\frac{du}{dx} = 15x^2.

Now, using the chain rule, we have:

\frac{dh}{dx} = \frac{1}{2\sqrt{u}} \cdot 15x^2

\frac{dh}{dx} = \frac{15x^2}{2\sqrt{5x^3 + 2}}.

4. Determine the derivative of k(x) = \cos(2x^2 + 3).

Solution:

To differentiate a function of a function, we can use the chain rule. Let u = 2x^2 + 3, then k(x) = \cos(u).

Now, let's differentiate u with respect to x:

\frac{du}{dx} = 4x.

Now, applying the chain rule, we get:

\frac{dk}{dx} = -\sin(u) \cdot 4x

\frac{dk}{dx} = -4x\sin(2x^2 + 3).

5. Find the derivative of m(x) = e^{x^2 + 2x + 3}.

Solution:

To differentiate a function of a function, we can use the chain rule. Let u = x^2 + 2x + 3, then m(x) = e^u.

Now, let's differentiate u with respect to x:

\frac{du}{dx} = 2x + 2.

Now, applying the chain rule, we have:

\frac{dm}{dx} = e^u \cdot (2x + 2)

\frac{dm}{dx} = (2x + 2)e^{x^2 + 2x + 3}.

Questions of Functions of Function

1. Find the derivative of the function f(x) = \sin(3x^2).

2. Calculate the derivative of g(x) = e^{2x^3}.

3. Determine the derivative of h(x) = \cos(4x).

4. Find the derivative of p(x) = \ln(5x^2 + 3).

5. Calculate the derivative of q(x) = \sqrt{2x + 1}.

6. Determine the derivative of r(x) = \tan(2x^2).

7. Find the derivative of s(x) = e^{\frac{1}{x^2}}.

8. Calculate the derivative of t(x) = \ln(4x^3 - 2x + 5).

9. Determine the derivative of u(x) = \sin(5x).

10. Find the derivative of v(x) = \cos(3x^2 + 2x).

11. Calculate the derivative of w(x) = e^{6x^2 + 3x}.

12. Determine the derivative of y(x) = \ln(2x+1)^2.

13. Find the derivative of z(x) = \sqrt{3x^2 + 1}.

14. Calculate the derivative of a(x) = \tan(4x^3 + x).

15. Determine the derivative of b(x) = e^{8x^2 - 2x}.

 

Corresponding Answers of Functions of Function

1. The derivative of f(x) = \sin(3x^2) is f'(x) = 6x\cos(3x^2).

2. The derivative of g(x) = e^{2x^3} is g'(x) = 6x^2e^{2x^3}.

3. The derivative of h(x) = \cos(4x) is h'(x) = -4\sin(4x).

4. The derivative of p(x) = \ln(5x^2 + 3) is p'(x) = \frac{10x}{5x^2 + 3}.

5. The derivative of q(x) = \sqrt{2x + 1} is q'(x) = \frac{1}{\sqrt{2x + 1}}.

6. The derivative of r(x) = \tan(2x^2) is r'(x) = 4x\sec^2(2x^2).

7. The derivative of s(x) = e^{\frac{1}{x^2}} is s'(x) = -\frac{2}{x^3}e^{\frac{1}{x^2}}.

8. The derivative of t(x) = \ln(4x^3 - 2x + 5) is t'(x) = \frac{12x^2 - 2}{4x^3 - 2x + 5}.

9. The derivative of u(x) = \sin(5x) is u'(x) = 5\cos(5x).

10. The derivative of v(x) = \cos(3x^2 + 2x) is v'(x) = -6x\sin(3x^2 + 2x) - 2\sin(3x^2 + 2x).

11. The derivative of w(x) = e^{6x^2 + 3x} is w'(x) = (12x + 3)e^{6x^2 + 3x}.

12. The derivative of y(x) = \ln(2x+1)^2 is y'(x) = \frac{2}{2x+1}.

13. The derivative of z(x) = \sqrt{3x^2 + 1} is z'(x) = \frac{3x}{\sqrt{3x^2 + 1}}.

14. The derivative of a(x) = \tan(4x^3 + x) is a'(x) = 12x^2\sec^2(4x^3 + x) + \sec^2(4x^3 + x).

15. The derivative of b(x) = e^{8x^2 - 2x} is b'(x) = (16x - 2)e^{8x^2 - 2x}.

Previous Lesson
Differential Calculus Playground
Next Lesson
Classroom: Differentiation of a quotient

Classroom: Differentiation function of a function

\frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx}

Calculus Archives

Useful Calculus links

Classroom: Differentiation function of a function
To differentiate a function of a function, also known as composite functions, you can use the chain rule....
Differentiate 3In5x With Steps
To differentiate the expression 3ln5x with respect to x, we will use the chain rule of differentiation....
Differentiate 3x With Steps
Learn how to differentiate the coefficient of 3x in this comprehensive guide. Brush up on your calculus...
Integral Calculus Solver AI
Integral calculus AI solver can help solve various types of integration problems, including standard...
Classroom: Differentiation of a quotient
The differentiation of a quotient involves applying the quotient rule, which states that the derivative...
Ads Blocker Image Powered by Code Help Pro

Ads Blocker Detected!!!

We have detected that you are using extensions to block ads. Please support us by disabling these ads blocker.

Powered By
100% Free SEO Tools - Tool Kits PRO