Derivative of Sec Square x
The derivative of sec square x is equal to twice the product of sec square x and tanx. Mathematically, we can write the derivative of sec^2x as d(sec^2x)/dx = 2 sec^{2}x tanx. We can evaluate the derivative of sec^2x using different methods of differentiation including the chain rule method, product rule and quotient rule of derivatives, and the first principle of derivatives, that is, the definition of limits. Differentiation of a function gives the slope function of the curve of the function which can give the slope of the function at a particular point.
Let us learn to evaluate the derivative of sec square x using different methods of derivatives and its formula. We will also solve different examples related to the concept for a better understanding of the concept of derivatives.
What is Derivative of Sec^2x?
The derivative of sec^2x is equal to 2 sec^{2}x tanx. It is mathematically written as d(sec^2x)/dx = 2 sec^{2}x tanx. Sec x is one of the important and main trigonometric functions in trigonometry. We know that the derivative of secx is equal to secx tanx. We can use the power rule of differentiation and the derivative of sec x to find the derivative of sec square x. In the next section, let us go through the formula for the differentiation of sec^2x.
Derivative of Sec Square x Formula
The formula for the derivative of sec^2x is given by d(sec^2x)/dx = 2 sec^2x tanx. We know that the power rule of differentiation is d(x^n)/dx = n x^(n1), that is, d(x^{n})/dx = nx^{n1}. Using this formula, we can write the formula for the differentiation of sec square x as d(sec^2x)/dx = 2 secx × d(secx)/dx = 2 secx × secx tanx = 2 sec^{2}x tanx. The image below shows the formula for the derivative of sec square x:
Derivative of Sec^2x Using Chain Rule
The chain rule of derivatives is used to find the derivatives of composite functions. Sec^2x is the composite function of two functions f(x) = x^{2} and g(x) = secx given by, h(x) = f o g (x) = sec^2x. Now, to find the derivative of sec^2x, we use the formula of chain rule given by, h'(x) = [f(g(x))]' = f'(g(x)) × g'(x). Hence, using this formula, the power rule of derivatives and the derivative of secx, we have
d(sec^2x)/dx = 2 sec^{21}x × d(secx)/dx
= 2 secx × secx tanx
= 2 sec^{2}x tanx
Hence, the derivative of sec square x is equal to 2 sec^{2}x tanx.
Differentiation of Sec Square Using Quotient Rule
Now that we know that the derivative of sec^2x is equal to 2 sec^{2}x tanx, we will prove this using the quotient rule of derivatives. We know that sec x is the reciprocal identity of cos x, that is, secx = 1/cosx. Also, the formula for the quotient rule of differentiation is (f/g)' = (f'g  fg')/g^{2}. Using these formulas, we have
d(sec^{2}x)/dx = d(1/cos^{2}x)/dx
= (1/cos^{2}x)'
= [(1)' cos^{2}x  1 (cos^{2}x)']/cos^{4}x
= (0 × cos^{2}x + 2 cosx sinx)/cos^{4}x  [Because the derivative of cosx is equal to sinx]
= (2 cosx sinx)/cos^{4}x
= 2 sinx/cos^{3}x
= 2 (1/cos^{2}x) (sinx/cosx)
= 2 sec^{2}x tanx
Hence, we have proved that the derivative of sec square is equal to 2 sec^{2}x tanx.
Derivative of Sec^2x Using Product Rule
The product rule of differentiation is used to find the derivative of the product of two functions. For a function h(x) = f(x) g(x), the formula for its derivative is given by h'(x) = f'(x) g(x) + f(x) g'(x). For the function sec square x, we can writ it as h(x) = sec^{2}x = secx × secx. Therefore, to find the derivative of sec^2x, we have
d(sec^{2}x)/dx = (secx × secx)'
= (secx)' secx + secx (secx)'
= secx tanx secx + secx secx tanx  [Because the derivative of secx is equal to secx tanx]
= 2 sec^{2}x tanx
Hence, the derivative of sec^2x is equal to 2 sec^{2}x tanx.
Important Notes on Derivative of Sec Square x
 The derivative of sec^2x is equal to 2 sec^{2}x tanx.
 The formula for the derivative of sec square x is d(sec^{2}x)/dx = 2 sec^{2}x tanx
 We can find the differentiation of sec square x using different methods of differentiation such as product rule, quotient rule, and chain rule.
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Derivative of Sec Square x Examples

Example 1: Determine the derivative of sec square x square with respect to x square.
Solution: To find the derivative of sec square x square with respect to x square, that is, d(sec^{2}(x^{2}))/d(x^{2}), we will use the chain rule method of derivatives.
Assume u = sec^{2}(x^{2}) and v = x^{2}. First, find the value of du/dx and dv/dx.
du/dx = d(sec^{2}(x^{2}))/dx
= 2 sec(x^{2}) × sec(x^{2}) tan(x^{2}) × 2x
= 4x sec^{2}(x^{2}) tan(x^{2})
dv/dx = d(x^{2})/dx
= 2x
To find the derivative of sec square x square with respect to x square, we need to find the value of du/dv. Therefore, we have
d(sec^{2}(x^{2}))/d(x^{2}) = du/dv
= [4x sec^{2}(x^{2}) tan(x^{2})]/2x
= 2 sec^{2}(x^{2}) tan(x^{2})
Answer: The derivative of sec^{2}(x^{2}) w.r.t. x^{2} is equal to 2 sec^{2}(x^{2}) tan(x^{2}).

Example 2: What is the second derivative of sec^2x?
Solution: To find the second derivative of sec^2x, we will differentiate its first derivative. We know that the derivative of sec^2x is equal to 2 sec^{2}x tanx. Differentiating this, we have
d^{2}(sec^{2}x)/dx^{2} = d(2 sec^{2}x tanx)/dx
= 2 [(sec^{2}x)' tanx + sec^{2}x (tanx)']  [Using product rule of differentiation]
= 2 [2 sec^{2}x tanx × tanx + sec^{2}x × sec^{2}x]  [Because the derivative of sec square x is 2 sec^{2}x tanx and the derivative of tanx is sec^{2}x]
= 2 [2 sec^{2}x tan^{2}x + sec^{4}x]
= 2 sec^{2}x [2 tan^{2}x + sec^{2}x]
Answer: The second derivative of sec square x is 2 sec^{2}x [2 tan^{2}x + sec^{2}x].

Example 3: Prove that the derivative of sec square x is 2 sec^{2}x tanx using the trigonometric identity 1 + tan^{2}x = sec^{2}x.
Solution: We know that 1 + tan^{2}x = sec^{2}x. Now, replacing sec^{2}x with 1 + tan^{2}x in the derivative, we have
d(sec^{2}x)/dx = d(1 + tan^{2}x)/dx
= d(1)/dx + d(tan^{2}x)/dx
= 0 + 2 tanx sec^{2}x  [Because derivative of tanx is sec^{2}x and d(x^{n})/dx = nx^{n1}]
= 2 sec^{2}x tanx
Answer: Hence Proved.
FAQs on Derivative of Sec^2x
What is Derivative of Sec Square x?
The derivative of sec square x is equal to twice the product of sec square x and tanx. Mathematically, we can write the derivative of sec^2x as d(sec^2x)/dx = 2 sec^{2}x tanx.
What is the Formula for the Derivative of Sec^2x?
The formula for the derivative of sec^2x is given by d(sec^{2}x)/dx = 2 sec^{2}x tanx. Using the formula for the power rule of derivatives, we can write the formula for the differentiation of sec square x as d(sec^2x)/dx = 2 secx × d(secx)/dx = 2 secx × secx tanx = 2 sec^{2}x tanx.
How Do You Prove the Derivative of Sec Square x?
We can evaluate the derivative of sec^2x using different methods of differentiation including the chain rule method, product rule and quotient rule of derivatives, and the first principle of derivatives, that is, the definition of limits.
What is the Derivative of Sec Square x Square w.r.t. x Square?
The derivative of sec square x square with respect to x square is equal to 2 sec^{2}(x^{2}) tan(x^{2}). We can find this using the chain rule method of differentiation.
How to Find the Second Derivative of Sec^2x?
To find the second derivative of sec^2x, we will differentiate its first derivative. We know that the derivative of sec^2x is equal to 2 sec^{2}x tanx. Differentiating this gives us the second derivative of sec square x. The second derivative of sec square x is 2 sec^{2}x [2 tan^{2}x + sec^{2}x].
What is the Antiderivative of Sec^2x?
The antiderivative of sec^2x is equal to tanx + C, where C is the integration constant. The antiderivative of a function is nothing but its integral and we know that the derivative of tanx is equal to sec^{2}x. Therefore, the antiderivative of sec square x is tanx + C.
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